tag:blogger.com,1999:blog-74344271938655784062024-03-13T21:38:34.611-07:00Coffee Cup Cardioids (And Other Diversions)One girl's adventures in mixing math with art. Calculus and geometry and fractals, doodles and sculpture and various craft projects (and more than a little coffee).Amyhttp://www.blogger.com/profile/14741075245934673435noreply@blogger.comBlogger30125tag:blogger.com,1999:blog-7434427193865578406.post-58456150789764762592014-03-28T00:18:00.002-07:002014-03-28T00:19:31.008-07:00Non-Euclidean Musings<iframe allowfullscreen="" frameborder="0" height="315" src="//www.youtube.com/embed/1AiK2khZ3lI?rel=0" width="560"></iframe><br />
<br />Amyhttp://www.blogger.com/profile/14741075245934673435noreply@blogger.com0tag:blogger.com,1999:blog-7434427193865578406.post-7736421939049367672013-11-07T19:15:00.000-08:002013-11-07T19:15:34.767-08:00Twist - Exploring the Cube and its Faces<div class="separator" style="clear: both; text-align: center;">
<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjJc3dnNA-Xv3KTGtKR_tiu1rmt_sTrCnZ0khJgacXI1eqobewd4ITnP4WtkuWgBSaiBbYdfOVsQT5789lzj8_EvVOiPHE11c7Biwn0wL4_xu-CN1Wo1nZQw8H27bIRGYR0qv8SjrMVU4w/s1600/IMG_3567.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="400" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjJc3dnNA-Xv3KTGtKR_tiu1rmt_sTrCnZ0khJgacXI1eqobewd4ITnP4WtkuWgBSaiBbYdfOVsQT5789lzj8_EvVOiPHE11c7Biwn0wL4_xu-CN1Wo1nZQw8H27bIRGYR0qv8SjrMVU4w/s400/IMG_3567.jpg" width="300" /></a></div>
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A recent sculptural undertaking of mine - although it may not seem like it at first, this tangled mess is actually based upon the cube. It's made of 24 empty pen barrels, tied together with rubber bands. There are six square faces, two of each color (the faces of the same color are parallel). Essentially, it is what I think would happen if you took a cube, pushed its square sides inward, and gave them a twist. Below is a single square face -<br />
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhKdF5k8OPbbtxvogg9TloJrzHgvuFHAhl-GoTxG33wSTZOVftV4CiAI2GSB_eWU2Ft5ZdOYmS3k3XRBx2tXkNl5GErQzegh-rNa9Fc88hL6-Qc6JAZWUIEeYARL23sJbj-P-gfBzHRy7c/s1600/IMG_3569+2.JPG" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="320" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhKdF5k8OPbbtxvogg9TloJrzHgvuFHAhl-GoTxG33wSTZOVftV4CiAI2GSB_eWU2Ft5ZdOYmS3k3XRBx2tXkNl5GErQzegh-rNa9Fc88hL6-Qc6JAZWUIEeYARL23sJbj-P-gfBzHRy7c/s320/IMG_3569+2.JPG" width="313" /></a></div>
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Then, each of the three pairs of faces are arranged with a <span style="background-color: white; color: #222222; font-family: arial, sans-serif; font-size: 13px; line-height: 16px;">π/4 </span>twist, which looks like this when viewed face on -<br />
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEj-WYtk9rPVZOl6zYOIT9KoGGdHNUXQ1TpdRjjNBqSsrDWOziWr_h3nGKbqHBjIDhXPxgthRkAQ314XwoUfeIERIQgC_kIit4VPc1FLkx0O7YOuFPYl0nuuj3y65fbWSv4ApBwlosnCDu4/s1600/IMG_3569.JPG" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="320" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEj-WYtk9rPVZOl6zYOIT9KoGGdHNUXQ1TpdRjjNBqSsrDWOziWr_h3nGKbqHBjIDhXPxgthRkAQ314XwoUfeIERIQgC_kIit4VPc1FLkx0O7YOuFPYl0nuuj3y65fbWSv4ApBwlosnCDu4/s320/IMG_3569.JPG" width="313" /></a></div>
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This arrangement makes eight convenient slots where the other four faces can interlock, as shown here. The darker/opaque green and purple dots indicate how the green and purple squares fit into the blue ones. Imagine that the set of green faces and the set of purple faces both also have a half twist to them. Also, the lighter green and purple dots on the intersection of the green and purple pieces drawn indicate the corners of the faces (the green dots on the purple pieces means the green overlaps there and two green sides come together at that point as a corner, and vice versa for the purple dots on the green pieces). You can actually see this a bit in the first picture of the actual sculpture, where the green and blue faces are held in place by the purple, and how the corners of the blue and green meet up and overlap. <br />
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjinPhiZlndaARtq-KNWMzL3dCNizGXODMU8rYwkVb3lEYcxmpRWRv5rhRMTQH2Tqka_JmU_h7R8QKyL4oUkVdYYgdydSe8yVKQ-kB1kLZBU2Qz5nTiYhTWa_3ga479K64YawcTInTKmyU/s1600/IMG_35691.JPG" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="320" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjinPhiZlndaARtq-KNWMzL3dCNizGXODMU8rYwkVb3lEYcxmpRWRv5rhRMTQH2Tqka_JmU_h7R8QKyL4oUkVdYYgdydSe8yVKQ-kB1kLZBU2Qz5nTiYhTWa_3ga479K64YawcTInTKmyU/s320/IMG_35691.JPG" width="313" /></a></div>
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If you turned the piece around, you would see this arrangement three times, once with each color serving as the outer "framework" as the blue is shown here. This was actually quite challenging to assemble, as it became a feat of intense coordination to hold everything together before it was all tied in place, but finished up, it pretty much holds itself together now. I made each of the squares smaller to tighten up the structure and I like how the ends sticking out at each corner give the whole thing a messier look - a curious juxtaposition of order and chaos.Amyhttp://www.blogger.com/profile/14741075245934673435noreply@blogger.com0tag:blogger.com,1999:blog-7434427193865578406.post-48455257141311068252013-11-02T15:08:00.000-07:002013-11-02T15:08:33.509-07:00Borromean Glow Stick Bracelets As a (slightly belated) Halloween post, here's a cool thing you can do with glow stick bracelets (or really anything, I just happen to have a lot of these now) -<br />
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEipUfbFfncXyijNpS4c4Gi49wmN064WM0cw4aM-sEapwjPOlQEgDg7ZQDgvLZenmSqSL7Q_aK_V-0dIAERBP9WyNt-tb7W5vNI_M_cccZjWpOVUOfH7ScvbSw8Qj4JQ1RUVwj_zXifIbVg/s1600/IMG_3504.JPG" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEipUfbFfncXyijNpS4c4Gi49wmN064WM0cw4aM-sEapwjPOlQEgDg7ZQDgvLZenmSqSL7Q_aK_V-0dIAERBP9WyNt-tb7W5vNI_M_cccZjWpOVUOfH7ScvbSw8Qj4JQ1RUVwj_zXifIbVg/s1600/IMG_3504.JPG" height="400" width="398" /></a></div>
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Check it out! This arrangement is known as the Borromean rings and is particularly interesting because no two rings are actually linked to each other, yet all three together are linked. They can be worn like so in a nice, mathematically interesting bracelet. <br />
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgstdS36puRWJxWmAb0EAXCklE-wkBXmOEiLtXP6WZmLqM_pjVPy3Uq6nJehbY_L9eBpDya-nNQhYqOlPHSxX7dOuBTQp11ThM4ixfvtnIFVidQ8SV9BMl-o1SZGvsYQn3U8Bmaa6vY4Lw/s1600/IMG_3505.JPG" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgstdS36puRWJxWmAb0EAXCklE-wkBXmOEiLtXP6WZmLqM_pjVPy3Uq6nJehbY_L9eBpDya-nNQhYqOlPHSxX7dOuBTQp11ThM4ixfvtnIFVidQ8SV9BMl-o1SZGvsYQn3U8Bmaa6vY4Lw/s1600/IMG_3505.JPG" height="273" width="400" /></a></div>
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Putting these rings together is fantastically simple. First take two bracelets (or other circular objects of your choice) and overlap one on the other like so. </div>
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Then, take the third bracelet and weave it through the other two, alternating over and under as you come around. </div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjEh8pH5Bw8SJQ3dDInrgYsUz1e7mNr14cCze2eJ3rRoEw4FFICTqRFQ_oFo4hQFCRBHxfmDPUcboTsNNJ78SJfxd5nhPdUbJe6ZivpS6RXmrC1b97hah6MqiP0xnh4qkfz-RKVPBRWZEU/s1600/IMG_3512.JPG" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjEh8pH5Bw8SJQ3dDInrgYsUz1e7mNr14cCze2eJ3rRoEw4FFICTqRFQ_oFo4hQFCRBHxfmDPUcboTsNNJ78SJfxd5nhPdUbJe6ZivpS6RXmrC1b97hah6MqiP0xnh4qkfz-RKVPBRWZEU/s1600/IMG_3512.JPG" height="240" width="320" /></a></div>
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Then simply connect up the two ends of the third bracelet and voila! </div>
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If you look closely, you'll find some other neat properties. For example, if you were to cut the Borromean rings, you would get one iteration of the standard three-strand braid, suggesting an alternate method of construction. Just as if you were to remove one strand of a braid and cause it to fall apart, you can try and take out one ring and see that the other two are no longer linked. </div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgNIVo-a5yP64Oj1YWSlTixz2VM6T99sRCI1vson1f2GCEnRI4Oa-jsYwbRz0lfv4PuBw7FDuSPBzQujwm9hgPKDNaAexr8Nq65oTO8dqA8kBaiMLy2ZSOVJvhWsofzSGIQDfyMtb8_SyA/s1600/IMG_3510.JPG" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgNIVo-a5yP64Oj1YWSlTixz2VM6T99sRCI1vson1f2GCEnRI4Oa-jsYwbRz0lfv4PuBw7FDuSPBzQujwm9hgPKDNaAexr8Nq65oTO8dqA8kBaiMLy2ZSOVJvhWsofzSGIQDfyMtb8_SyA/s1600/IMG_3510.JPG" height="300" width="400" /></a></div>
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Also notice how if you look at any one ring, it is wholly inside of, and wholly outside of, the other two rings. <br />
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Alternatively, you could dip your rings into some <a href="http://coffeecupcardioids.blogspot.com/2013/08/optimal-bubbleology.html">bubble solution</a>. The result is one of a class of objects called Seifert surfaces, any surface defined by a knot or link. Here is a beautiful sculpture of this surface (<a href="http://www.bathsheba.com/math/borromean/">image source</a>) <br />
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And just think, all of this from three simple rings... </div>
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<span style="font-family: Georgia,"Times New Roman",serif;"><span style="font-size: large;"><b>"I'm just playing. That's what math is - wondering, playing, amusing yourself with your imagination."</b> - Paul Lockhart </span></span></div>
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Amyhttp://www.blogger.com/profile/14741075245934673435noreply@blogger.com0tag:blogger.com,1999:blog-7434427193865578406.post-41456612527218947402013-10-23T22:41:00.000-07:002013-10-23T22:41:37.032-07:00An Afterthought on Tessellations<div class="separator" style="clear: both; text-align: center;">
<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjZLlcN7LCMhG5J1b4xqCUueK3W8q0Am58L1A4l0iqoyUuBr_jHfEffeqquerFxHQAxhgI_BrBAjfNc3U1KE8x5PvrWl67bsBK5JMsBPcdNuyC5CEiYSkPa_8rX_RfNOCP1RuIK22wt4zs/s1600/IMG_3371.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="361" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjZLlcN7LCMhG5J1b4xqCUueK3W8q0Am58L1A4l0iqoyUuBr_jHfEffeqquerFxHQAxhgI_BrBAjfNc3U1KE8x5PvrWl67bsBK5JMsBPcdNuyC5CEiYSkPa_8rX_RfNOCP1RuIK22wt4zs/s400/IMG_3371.jpg" width="400" /></a></div>
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Check out this awesome tiling I saw today - this place had its floor tiles cut like Escher's lizard tessellations! </div>
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The four types of tessellations - <a href="http://coffeecupcardioids.blogspot.com/2013/09/twisted-tessellations-i-translation.html">translations</a>, <a href="http://coffeecupcardioids.blogspot.com/2013/09/twisted-tessellations-ii-reflection.html">reflections</a>, <a href="http://coffeecupcardioids.blogspot.com/2013/10/twisted-tessellations-iii-rotation.html">rotations</a>, and <a href="http://coffeecupcardioids.blogspot.com/2013/10/twisted-tessellations-iv-glide.html">glide reflections</a></div>
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Amyhttp://www.blogger.com/profile/14741075245934673435noreply@blogger.com0tag:blogger.com,1999:blog-7434427193865578406.post-51198721969271164932013-10-22T22:40:00.000-07:002013-11-02T15:23:14.499-07:00Food for ThoughtHere's a thought - take a slice of cheese. With n number of cuts, what is the maximum number of pieces you could cut your cheese into? Observe...<br />
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiQjVcN9TKUwtJSRBwfzxUSJ8p-vTD4vPV2x408QL8IeTU3FMuaSdozMDWrVTa7IEgn5s9Bi7STOXivC4ECmJJrVokxT35ghOdr84Q2kKB2hdM2qiGAeMQH1HKnakAI10AMwenM8Su-KC0/s1600/IMG_2138.JPG" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="320" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiQjVcN9TKUwtJSRBwfzxUSJ8p-vTD4vPV2x408QL8IeTU3FMuaSdozMDWrVTa7IEgn5s9Bi7STOXivC4ECmJJrVokxT35ghOdr84Q2kKB2hdM2qiGAeMQH1HKnakAI10AMwenM8Su-KC0/s320/IMG_2138.JPG" width="298" /></a></div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEg7RQQfhcRum7vBUpfK06Gf_CIW4vwQZhg9AYtlSQl_Y2VDFqYC_xRownZhJJybDHx7qBMPwIwkdDOQOc5LBehoyjNJKikKInFNokN5l-wSu_Ly_lF9It_NZgenJ-cYfLK1vk6esDubxEw/s1600/IMG_2139.JPG" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="320" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEg7RQQfhcRum7vBUpfK06Gf_CIW4vwQZhg9AYtlSQl_Y2VDFqYC_xRownZhJJybDHx7qBMPwIwkdDOQOc5LBehoyjNJKikKInFNokN5l-wSu_Ly_lF9It_NZgenJ-cYfLK1vk6esDubxEw/s320/IMG_2139.JPG" width="286" /></a></div>
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Notice how as we add each cut, starting at 0 cuts with 1 slice, we get 1, 2, 4, 7, 11... </div>
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A lovely pattern can be seen in the maximum number of pieces you get where you take the previous number and add 1, then 2, then 3, then 4... perhaps more familiarly as the triangular numbers + 1 or n(n+1)/2 + 1</div>
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Continuing with that thought, what would happen if you extended this problem to say, a block of cheese? (going from 2D to 3D). Given a certain number of planar cuts, what is the maximum number of pieces you can create? With 2D cheese, when adding a cut, to maximize the number of pieces you get, your line has to cut through each of the existing cuts/lines. The same is true in the case of 3D cheese, in that each new plane must cut all of the existing planes. The following pattern emerges: </div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjBnfgnPye7d8nPBf2qDR80kJ-USJX9KXeSpcpYVdApqmZWOTit7p4h226NK9M_N6tSOerACy0sVx3RSo3QHk6YMrKaIp0u97OJ0AUCRkC_VfoKeeqEvkyHTcLBYG_LawrDN25x3rAT3GU/s1600/IMG_2142.JPG" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="318" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjBnfgnPye7d8nPBf2qDR80kJ-USJX9KXeSpcpYVdApqmZWOTit7p4h226NK9M_N6tSOerACy0sVx3RSo3QHk6YMrKaIp0u97OJ0AUCRkC_VfoKeeqEvkyHTcLBYG_LawrDN25x3rAT3GU/s400/IMG_2142.JPG" width="400" /></a></div>
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Notice how with 2D cheese, each time you were adding 1, then 2, 3, etc. but then in 3D cheese, the numbers your adding each time are the numbers of pieces per cut from the 2D cheese problem. With the previous formula, you can derive the following pattern in 3D: (n^3+5n+6)/6<br />
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Additional notes: </div>
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<li style="text-align: left;">This is actually called the lazy caterer's sequence in 2D and cake numbers in 3D </li>
<li style="text-align: left;">It's pretty cool how for n dimensions, the first n slices will be consecutive powers of 2, and the next one will be 2^(n+1)-1</li>
<li style="text-align: left;">For any given number of slices, can you cut the 2D or 3D cheese into equally sized pieces? If so, how, and if not, why not? (I actually have no idea) </li>
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Amyhttp://www.blogger.com/profile/14741075245934673435noreply@blogger.com0tag:blogger.com,1999:blog-7434427193865578406.post-32343594731527651512013-10-07T21:02:00.001-07:002013-10-08T18:45:27.529-07:00Twisted Tessellations IV - Glide Reflection<div class="separator" style="clear: both; text-align: center;">
<a href="http://www.math.nus.edu.sg/aslaksen/gem-projects/maa/0203-2-03-Escher/horseman.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="238" src="http://www.math.nus.edu.sg/aslaksen/gem-projects/maa/0203-2-03-Escher/horseman.jpg" width="320" /></a></div>
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Original artwork by M. C. Escher - <a href="http://www.math.nus.edu.sg/aslaksen/gem-projects/maa/0203-2-03-Escher/main3.html">image source.</a> To finish off the series of tessellation posts (see <a href="http://coffeecupcardioids.blogspot.com/2013/09/twisted-tessellations-i-translation.html">translation</a>, <a href="http://coffeecupcardioids.blogspot.com/2013/09/twisted-tessellations-ii-reflection.html">reflection</a>, and <a href="http://coffeecupcardioids.blogspot.com/2013/10/twisted-tessellations-iii-rotation.html">rotation</a> for more), here's in my opinion the coolest one - glide reflections. It is, like it sounds, a combination of flipping and sliding; notice how here you could take one of the white knights, flip it, and slide it to get to a grey knight. </div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhq0X_UC3mWVvNd0eY3XZA7jifY6QUThBCe08n7dsggGYCzYixGvk_DaTPKINFNbbKWXu_TYxWd43G6mVHfPAHgEijoi1iAAUi26aaxwTVpY2Gc-3SEEflBNUEmG4Nl0mAy8BeZ0ckkfzU/s1600/IMG_3269.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="200" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhq0X_UC3mWVvNd0eY3XZA7jifY6QUThBCe08n7dsggGYCzYixGvk_DaTPKINFNbbKWXu_TYxWd43G6mVHfPAHgEijoi1iAAUi26aaxwTVpY2Gc-3SEEflBNUEmG4Nl0mAy8BeZ0ckkfzU/s200/IMG_3269.jpg" width="197" /></a><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiepxvl77-JAFgrzQzCzcVXsROcUlSStGfs0Wy5DEmHafLjKttSNeOS7ozRcZlyxRD5Q_ZxNviiQPPaucAHdfDjCR6B_elBRbWMbRClDzUXPPHDBpgEEd5zQW0Joed29z27F8wszzDspa0/s1600/IMG_3270.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="200" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiepxvl77-JAFgrzQzCzcVXsROcUlSStGfs0Wy5DEmHafLjKttSNeOS7ozRcZlyxRD5Q_ZxNviiQPPaucAHdfDjCR6B_elBRbWMbRClDzUXPPHDBpgEEd5zQW0Joed29z27F8wszzDspa0/s200/IMG_3270.jpg" width="181" /></a></div>
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<b>Step 1 </b>- Take your square, draw your usual squiggly bit on one side and cut out. </div>
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<b>Step 2 </b>- Flip the piece over and attach to the opposite side. </div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjR_SN1Q_Qh7WjZ86MZ6-ScTOrmj6yzaoyD6WcGrx9PYsmyMBJwMdFP0VN-c5kG9wlnsdZUtE2wmnA9DyGLab09Z5gG85ucOziUkvWfF5hgsXh8VVdarKuIZeUez3WFk3JG2pVFwwvOUdY/s1600/IMG_3271.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="200" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjR_SN1Q_Qh7WjZ86MZ6-ScTOrmj6yzaoyD6WcGrx9PYsmyMBJwMdFP0VN-c5kG9wlnsdZUtE2wmnA9DyGLab09Z5gG85ucOziUkvWfF5hgsXh8VVdarKuIZeUez3WFk3JG2pVFwwvOUdY/s200/IMG_3271.jpg" width="176" /></a><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiH3-Kz5dyyzL5CJ_vRHoyEy4Sw6ydYAi808Kotdvnr2qDnL26jTO55-tkExPo3Vu-jvTVFA0IW26hYb0HdZHFuI_JlFlTdnb2XWoJk4TQrGBFh5HdL0tv8YWPqwRB18dDZzFupL9BHzEE/s1600/IMG_3272.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="176" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiH3-Kz5dyyzL5CJ_vRHoyEy4Sw6ydYAi808Kotdvnr2qDnL26jTO55-tkExPo3Vu-jvTVFA0IW26hYb0HdZHFuI_JlFlTdnb2XWoJk4TQrGBFh5HdL0tv8YWPqwRB18dDZzFupL9BHzEE/s200/IMG_3272.jpg" width="200" /></a></div>
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<b>Step 3 </b>- Choose one of the two untouched sides, draw another squiggle and cut out. </div>
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<b>Step 4 </b>- Flip it over and attach to the opposite side like you did with the first piece in Step 2. Now your lovely tessellation is ready to trace! </div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiN6bwaCdqRBUBQ1LJTLcQbLfb3xDf2DP9CdNq9IN4xcBQjeVFbJSyXdUrLXWiEAkfV9mURemhDhbLZ9YHOcWAfhcwYVnMlVhEqv9oJwLe7SvQ2xxrXfrcDbT-a5QO5D7bvy4j12XErFFQ/s1600/IMG_3273.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="400" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiN6bwaCdqRBUBQ1LJTLcQbLfb3xDf2DP9CdNq9IN4xcBQjeVFbJSyXdUrLXWiEAkfV9mURemhDhbLZ9YHOcWAfhcwYVnMlVhEqv9oJwLe7SvQ2xxrXfrcDbT-a5QO5D7bvy4j12XErFFQ/s400/IMG_3273.jpg" width="381" /></a></div>
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This one is possibly the hardest of the four to trace because it involves a lot of flipping and fitting squiggly bits together, but if you chose to use paper that has different colors on each side, you can think when you traced the first shape (see the picture for Step 4), the top edge was the red edge with the triangle poking out, so when you flip the piece over, you'd align the piece so the red edge with the space for the triangle would be on the bottom. </div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiGlvzcnudBszivcEjpEDsPUNSriPHk4mVpcokgc2_aTqLvI_R7RtXWtyam3pp5wV-saOX4gPbxzB_XXG4TVcQiuE4SvofZiIjMwmHWgvnn-2tL0eNZQa3RkDbebg_POjcIVptCkqJmdH8/s1600/IMG_3274.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="342" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiGlvzcnudBszivcEjpEDsPUNSriPHk4mVpcokgc2_aTqLvI_R7RtXWtyam3pp5wV-saOX4gPbxzB_XXG4TVcQiuE4SvofZiIjMwmHWgvnn-2tL0eNZQa3RkDbebg_POjcIVptCkqJmdH8/s400/IMG_3274.jpg" width="400" /></a></div>
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It takes a bit of practice, but the results are quite pretty! </div>
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Further exploration - </div>
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<ul>
<li>I happened to use squares as the base shape, but you can definitely use others too - triangles are nice, so are rhombuses. </li>
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<li>That said, can you start with any shape? </li>
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<li>These tessellations were all done with only one shape, but can you do something with two, three, four, etc. shapes? </li>
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Amyhttp://www.blogger.com/profile/14741075245934673435noreply@blogger.com0tag:blogger.com,1999:blog-7434427193865578406.post-69662602485727163792013-10-07T00:11:00.001-07:002013-10-07T00:12:25.587-07:00Mathspiration <iframe width="560" height="315" src="//www.youtube.com/embed/cIpSrAgY3Ng?rel=0" frameborder="0" allowfullscreen></iframe>
Basically, I found this tonight and it deserves way more views than it has. It's pretty awesome.Amyhttp://www.blogger.com/profile/14741075245934673435noreply@blogger.com1tag:blogger.com,1999:blog-7434427193865578406.post-36877857233557078872013-10-01T20:28:00.000-07:002013-10-08T18:45:36.501-07:00Twisted Tessellations III - Rotation <div class="separator" style="clear: both; text-align: center;">
<a href="http://britton.disted.camosun.bc.ca/tess104.GIF" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="313" src="http://britton.disted.camosun.bc.ca/tess104.GIF" width="320" /></a></div>
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Original artwork by M. C. Escher (can you tell I have an Escher obsession?) <a href="http://britton.disted.camosun.bc.ca/jbtess104.htm">Image source.</a> A third type of tessellation (in addition to <a href="http://coffeecupcardioids.blogspot.com/2013/09/twisted-tessellations-i-translation.html">translations</a> and <a href="http://coffeecupcardioids.blogspot.com/2013/09/twisted-tessellations-ii-reflection.html">reflections</a>) is rotation, where instead of sliding or flipping the pattern, you spin it about a pivot point. </div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEi3Wsq__7CBmYt0x3k6lUF-TFNCzL1k0RWJRexrM0F0LdrF8TxoSCgk5xhvfWq6BdbKWfQc7phY1hTCiXNXMMIgTuFw1Xq0uJGHyBu2qGbcvImJoT5QUiZiI-ADlfbDcABlx55idKlhSQU/s1600/IMG_3264.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="198" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEi3Wsq__7CBmYt0x3k6lUF-TFNCzL1k0RWJRexrM0F0LdrF8TxoSCgk5xhvfWq6BdbKWfQc7phY1hTCiXNXMMIgTuFw1Xq0uJGHyBu2qGbcvImJoT5QUiZiI-ADlfbDcABlx55idKlhSQU/s200/IMG_3264.jpg" width="200" /></a><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhTQQ6tzqXSrQFwuL8NCxZkGT_iMycvdpyCvPPGUT8sIbo3RY5oUkmebAYa2GWt5Q-rPCoe20mtKmo411IE6Gq2IFcSLd17J2RIE1_SxqEVOCgK1MsYvKtzx8Ovzy_rPcDhW7Csn0x07B4/s1600/IMG_3265.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="198" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhTQQ6tzqXSrQFwuL8NCxZkGT_iMycvdpyCvPPGUT8sIbo3RY5oUkmebAYa2GWt5Q-rPCoe20mtKmo411IE6Gq2IFcSLd17J2RIE1_SxqEVOCgK1MsYvKtzx8Ovzy_rPcDhW7Csn0x07B4/s200/IMG_3265.jpg" width="200" /></a></div>
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<b>Step 1</b> - Starting again with a square, mark off the midpoint on one of the edges, and draw and cut out any pattern or squiggle you'd like, starting at either corner and going to the midpoint. </div>
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<b>Step 2 </b>- Give the square a quarter turn and, using the piece you just cut out as a stencil, trace and cut out the same piece from the new side, making sure you stay consistent with the first edge (i.e. if you started cutting from the left corner, don't switch to starting from the right corner). Repeat with the remaining two sides.</div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhf9PrIwy7FO_kq-T5v1dUgkwG9NkER44EuowZdbbdrhdl9bN-UvfQoG0Y_bmSXnL2H0RyNVA9xTwdgXvG6Xa7iW9tCHqSCMsa0q0sjF5YfrmMMAlsq-4pSgCs4k6sjkZzqEqmwIUf2Ie8/s1600/IMG_3266.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="199" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhf9PrIwy7FO_kq-T5v1dUgkwG9NkER44EuowZdbbdrhdl9bN-UvfQoG0Y_bmSXnL2H0RyNVA9xTwdgXvG6Xa7iW9tCHqSCMsa0q0sjF5YfrmMMAlsq-4pSgCs4k6sjkZzqEqmwIUf2Ie8/s200/IMG_3266.jpg" width="200" /></a><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjJOQr2uLV8h0jFsECw8vjSPIRHhhJr6LJ7fzgI337OxscdQhxOHqA-vwU-fh-yqtx7hCPGUNBYGeWeAoVO-1dku-NbQvbwlCYsh95QpqAnPLNmw96YrGqZUIUvK5-YbOxgcv2m4V5Qf4c/s1600/IMG_3267.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="200" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjJOQr2uLV8h0jFsECw8vjSPIRHhhJr6LJ7fzgI337OxscdQhxOHqA-vwU-fh-yqtx7hCPGUNBYGeWeAoVO-1dku-NbQvbwlCYsh95QpqAnPLNmw96YrGqZUIUvK5-YbOxgcv2m4V5Qf4c/s200/IMG_3267.jpg" width="199" /></a></div>
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<b>Step 3 </b>- Now you should have four identical squiggly pieces cut out. On each edge of the square, there is one half that wasn't cut - position the four pieces on the uncut half as shown. If you've done it correctly, you should be able to hold the midpoint and pivot the piece about that point until it fits back into where it was cut from. </div>
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<b>Step 4 </b>- Tape the pieces accordingly and now your shape is ready to trace!</div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiB2GysOEKQ8G9v70amCQ3tnBmfMjgLQ_r3TjwRo77DvsN2-CKgCCkzKJtN2JgTljcMHnPgyE1qKpj52HWQsAcSyuHEQVd-zAnaWbGggUAiPMdOfakqgjciBvbfspfh6hSGyettiekWbSs/s1600/IMG_3268.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="271" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiB2GysOEKQ8G9v70amCQ3tnBmfMjgLQ_r3TjwRo77DvsN2-CKgCCkzKJtN2JgTljcMHnPgyE1qKpj52HWQsAcSyuHEQVd-zAnaWbGggUAiPMdOfakqgjciBvbfspfh6hSGyettiekWbSs/s320/IMG_3268.jpg" width="320" /></a></div>
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Can you see how this tessellation is both a rotation and a translation? </div>
Amyhttp://www.blogger.com/profile/14741075245934673435noreply@blogger.com0tag:blogger.com,1999:blog-7434427193865578406.post-91560058495135145232013-09-22T18:43:00.001-07:002013-10-08T18:45:47.639-07:00Triquetras, Trefoils, and Topological Awesomeness<div class="separator" style="clear: both; text-align: center;">
<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEh_PMwfBn1m1csUljHW2w6MRhDJeA54_StTcj1Yl3sqoCZyMjW98JI4u1kBdq1KJJXM8IMk-PTrGCVPu33VbB4FdtAe5ED8ARXaCXNG-_HB_PM-NXBQimrKbrIeec_5oRlIRBcVTRh8BT0/s1600/IMG_3327.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="348" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEh_PMwfBn1m1csUljHW2w6MRhDJeA54_StTcj1Yl3sqoCZyMjW98JI4u1kBdq1KJJXM8IMk-PTrGCVPu33VbB4FdtAe5ED8ARXaCXNG-_HB_PM-NXBQimrKbrIeec_5oRlIRBcVTRh8BT0/s400/IMG_3327.jpg" width="400" /></a></div>
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Here is a pretty awesome piece of mathematically interesting art I saw the other day. </div>
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<a href="http://upload.wikimedia.org/wikipedia/commons/6/65/Triquetra-Vesica.svg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="302" src="http://upload.wikimedia.org/wikipedia/commons/6/65/Triquetra-Vesica.svg" width="320" /></a></div>
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This shape is called a triquetra. The name comes from the Latin <i>tri-</i> meaning "three" and <i>quetrus </i>meaning "cornered." <a href="http://en.wikipedia.org/wiki/Triquetra"></a></div>
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<a href="http://upload.wikimedia.org/wikipedia/commons/b/b3/Blue_Trefoil_Knot.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="320" src="http://upload.wikimedia.org/wikipedia/commons/b/b3/Blue_Trefoil_Knot.png" width="300" /></a></div>
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It is analogous to the trefoil knot - what you get when you tie a simple overhand knot and join the two ends. </div>
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<a href="http://mathworld.wolfram.com/images/eps-gif/Triquetra_1000.gif" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="320" src="http://mathworld.wolfram.com/images/eps-gif/Triquetra_1000.gif" width="320" /></a></div>
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Another way to find a triquetra is to look at the intersection of three circles, like a Venn diagram. (Note that the shape formed at the very center is a Reuleaux triangle, a shape of constant width, deserving of a blog post all of its own...another day)</div>
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<a href="http://upload.wikimedia.org/wikipedia/commons/thumb/8/80/Triquetra-circle-interlaced.svg/528px-Triquetra-circle-interlaced.svg.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="310" src="http://upload.wikimedia.org/wikipedia/commons/thumb/8/80/Triquetra-circle-interlaced.svg/528px-Triquetra-circle-interlaced.svg.png" width="320" /></a></div>
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The triquetra is often shown interlaced with a circle and is common to Celtic art. It has many meanings throughout various religions and symbolizes things that are threefold - mind, body, soul; past, present, future; Father, Son, Holy Spirit; to name a few. </div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgabVpcUH7Jn6QB_3-gHoVuvAIw9bOOGeODESwckqbyeJAEPzwO92_WisWVp8cy3o8GVRWDHwdIzpiHaNXtY0DI3flXwL2KD4j_scKqZn7_M79OaTmEKx4Ebsl1IGq7-vgQutUe1XgLHtM/s1600/IMG_3329.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="286" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgabVpcUH7Jn6QB_3-gHoVuvAIw9bOOGeODESwckqbyeJAEPzwO92_WisWVp8cy3o8GVRWDHwdIzpiHaNXtY0DI3flXwL2KD4j_scKqZn7_M79OaTmEKx4Ebsl1IGq7-vgQutUe1XgLHtM/s400/IMG_3329.jpg" width="400" /></a></div>
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And if that wasn't cool enough, here's another cool thing you can do - take a strip of paper, give it three half-twists, and join the two ends into a three twist Möbius strip. </div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiVUPA4YPeNonTqxavSNogecxs-naPPgXpILhKlJnLEjzABfkRmhmkIEzQMZc87rlHSXpYZ5MugZDtpTvPbAv4TJ4wza8xpIdIxlnCHPrseeNx3_i_W6aOVzfqMP0gMCWm85JHIaRDQldg/s1600/IMG_3330.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="300" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiVUPA4YPeNonTqxavSNogecxs-naPPgXpILhKlJnLEjzABfkRmhmkIEzQMZc87rlHSXpYZ5MugZDtpTvPbAv4TJ4wza8xpIdIxlnCHPrseeNx3_i_W6aOVzfqMP0gMCWm85JHIaRDQldg/s400/IMG_3330.jpg" width="400" /></a></div>
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Then, cut the strip in half along the middle of the strip. After some rearranging and playing with the twists, BAM! Trefoil! How awesome is that? </div>
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For further exploration: </div>
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<ul>
<li>Draw a triquetra - <a href="http://www.had2know.com/makeit/draw-trinity-knot-triquetra.html">link</a> </li>
<li>Animation of Möbius strip to trefoil knot - <a href="http://www.youtube.com/watch?v=Ock9x-oD-7Y">link</a></li>
<li>George Hart shows how to cut a bagel into a trefoil knot for some "mathematically correct breakfast" - <a href="http://georgehart.com/bagel/knot.html">link</a> </li>
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Images 2 and 5 from <a href="http://en.wikipedia.org/wiki/Triquetra">here</a>, Image 3 from <a href="http://en.wikipedia.org/wiki/Trefoil_knot">here</a>, and Image 4 from <a href="http://mathworld.wolfram.com/Triquetra.html">here</a>.</div>
Amyhttp://www.blogger.com/profile/14741075245934673435noreply@blogger.com0tag:blogger.com,1999:blog-7434427193865578406.post-26248396198442504812013-09-15T21:21:00.000-07:002013-09-15T21:21:19.339-07:00Mathematical Broccoli <div class="separator" style="clear: both; text-align: center;">
<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhYLyxUueVHOaONmpKmxKbIb6uEoi_cJd93JoSfSRDmYZKeoGHnkuzraC53mbZ9TqQa3uzJRLNQgWBxs6o4HCvDVILoG04iJKvUaPZvEyc0V5yQYu31wIK6RLictFmBxbKOzIhKYdn8Dds/s1600/IMG_3282.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="400" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhYLyxUueVHOaONmpKmxKbIb6uEoi_cJd93JoSfSRDmYZKeoGHnkuzraC53mbZ9TqQa3uzJRLNQgWBxs6o4HCvDVILoG04iJKvUaPZvEyc0V5yQYu31wIK6RLictFmBxbKOzIhKYdn8Dds/s400/IMG_3282.jpg" width="387" /></a></div>
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This is hands down my new favorite vegetable. It's called Romanesco broccoli and I'd seen pictures of it online (it's also mentioned in Benoit Mandelbrot's TED talk <a href="http://www.youtube.com/watch?v=ay8OMOsf6AQ">here</a>) but never in person until tonight. Isn't it just gorgeous though? It approximates a fractal in that it is a pattern that repeats itself infinitely...or well, as far as the physical limits of broccoli allow, that is. If you were to cut off one of the little florets, it would look like a copy of the whole thing, but smaller. Also of interest is that the little buds are arranged in logarithmic spirals, and like <a href="http://coffeecupcardioids.blogspot.com/2013/09/fibonnaci-pinecone.html">pinecones</a>, counting the number of spirals going one way and then going the other way results in adjacent Fibonacci numbers. </div>
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Seriously, can we just admire how intricate and awesome this broccoli is?!<br />
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<a href="http://www.wired.com/wiredscience/wp-content/gallery/fractal/fractal_10.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="300" src="http://www.wired.com/wiredscience/wp-content/gallery/fractal/fractal_10.jpg" width="400" /></a></div>
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<a href="http://www.wired.com/wiredscience/2010/09/fractal-patterns-in-nature/">Image source.</a></div>
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<br />Amyhttp://www.blogger.com/profile/14741075245934673435noreply@blogger.com0tag:blogger.com,1999:blog-7434427193865578406.post-66801957582336502202013-09-15T09:53:00.000-07:002013-10-08T18:46:08.721-07:00Twisted Tessellations II - Reflection<div class="separator" style="clear: both; text-align: center;">
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<a href="http://www.math.nus.edu.sg/aslaksen/gem-projects/maa/0203-2-03-Escher/angels&devils.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="303" src="http://www.math.nus.edu.sg/aslaksen/gem-projects/maa/0203-2-03-Escher/angels&devils.jpg" width="320" /></a></div>
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Original artwork by M. C. Escher. <a href="http://www.math.nus.edu.sg/aslaksen/gem-projects/maa/0203-2-03-Escher/main3.html">Image source.</a> In addition to <a href="http://coffeecupcardioids.blogspot.com/2013/09/twisted-tessellations-i-translation.html">translations</a>, another type of tessellation is a reflection, where a shape has been flipped about either the x or y axis. </div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgNkB6kLPm6PhPeMxK6yaw3j5CgPDykpqjiTXDxtKgwWIrbAZq1eGBx7Ufb3O-T4cvxB4VYI0wV4eZxZLytkHFbwwzrxdG6NlmZW-nAD40Q1ADlI2S68sMMlPQCDwDpYvZ2NHLhsnWrRrw/s1600/IMG_3256.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="195" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgNkB6kLPm6PhPeMxK6yaw3j5CgPDykpqjiTXDxtKgwWIrbAZq1eGBx7Ufb3O-T4cvxB4VYI0wV4eZxZLytkHFbwwzrxdG6NlmZW-nAD40Q1ADlI2S68sMMlPQCDwDpYvZ2NHLhsnWrRrw/s200/IMG_3256.jpg" width="200" /></a><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEif4kyN2kt7iEATAzXQUySEGixvD33diwMlcUMoQ1HX2GaMdMIWpW5YX9MfFBUxJ4WXaPxpYrz1RwUAmY6uP0MF3-SIq51YcrVPuHG4FeqiDRnps9FEB9q6X1HuPrF38taSpF1zSdt-YK0/s1600/IMG_3257.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="200" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEif4kyN2kt7iEATAzXQUySEGixvD33diwMlcUMoQ1HX2GaMdMIWpW5YX9MfFBUxJ4WXaPxpYrz1RwUAmY6uP0MF3-SIq51YcrVPuHG4FeqiDRnps9FEB9q6X1HuPrF38taSpF1zSdt-YK0/s200/IMG_3257.jpg" width="197" /></a></div>
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<b>Step 1 </b>- Like for translations, start with a square piece of paper. Now pick one edge, mark off the midpoint, and draw some kind of squiggle from the corner to the center point. </div>
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<b>Step 2 </b>- Cut out the piece you've just drawn, flip it over, and line it up with the other half of the chosen edge. Trace and cut out. Now you'll have two little pieces that are reflections of each other. </div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEihc99d6sr1a5ayIy6PFG9R6bXC2nhggyho9WQ6HcWB7PGEBkzQOjA4KCOSI_vnk0oySuNVwOTMUoA7AhQeHWHKEw7Fblj1y3qwUvWdSxItIjhg7YHo9-63VwvyGNZMcYLLSB4luD9woEo/s1600/IMG_3258.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="196" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEihc99d6sr1a5ayIy6PFG9R6bXC2nhggyho9WQ6HcWB7PGEBkzQOjA4KCOSI_vnk0oySuNVwOTMUoA7AhQeHWHKEw7Fblj1y3qwUvWdSxItIjhg7YHo9-63VwvyGNZMcYLLSB4luD9woEo/s200/IMG_3258.jpg" width="200" /></a><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEh1N691BjiLAIdWOcE3jLDM-TTMhPikBHeu7SXYl_1XLpRQVRHPrr8ILLQP3yU1GFfh41oh_mpBtGOLAow9Rj0kWYhXuciCXFx0XRpD_k2c0gd02WcyXKtQSSBcDQO2kja6m-qdZdVF-NA/s1600/IMG_3259.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="145" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEh1N691BjiLAIdWOcE3jLDM-TTMhPikBHeu7SXYl_1XLpRQVRHPrr8ILLQP3yU1GFfh41oh_mpBtGOLAow9Rj0kWYhXuciCXFx0XRpD_k2c0gd02WcyXKtQSSBcDQO2kja6m-qdZdVF-NA/s200/IMG_3259.jpg" width="200" /></a></div>
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<b>Step 3 </b>- Take those two pieces and line them up on the opposite edge as shown. Trace and cut out.</div>
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<b>Step 4 </b>- Attach the four pieces you've cut out to the two uncut edges like so. If you've done this correctly, you'll be able to draw a line right down the middle both horizontally and vertically and have the two sides be mirror images of one another. </div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEje0gPfOnvnjtOCCvC86L3uS3aaMzwaSZ92FwXy0JOVFPZ00vSLVc5vkcBqbOqIhHM9n4riBVraF922h1HLiEEyEyy1aEGVixhbpSr1wXKhX7Qj8RfjR7z94czBbCLWxMQyaoK31f3QYos/s1600/IMG_3260.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="303" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEje0gPfOnvnjtOCCvC86L3uS3aaMzwaSZ92FwXy0JOVFPZ00vSLVc5vkcBqbOqIhHM9n4riBVraF922h1HLiEEyEyy1aEGVixhbpSr1wXKhX7Qj8RfjR7z94czBbCLWxMQyaoK31f3QYos/s400/IMG_3260.jpg" width="400" /></a></div>
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Now your beautiful stencil is ready to trace! </div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhKe-07J6pDzL2IiJufX7grIB8ecsKM0KKwQdAgIp0gfk4CXDD1GUC0PLBdTJCDJJg-E_ZrmmbdlF2kw8vJqw885nMDYxzxoC2UaUZNr0QQP8vIo68nRZEW-337AkqxgJNiaKggH4SMmYg/s1600/IMG_3261.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="273" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhKe-07J6pDzL2IiJufX7grIB8ecsKM0KKwQdAgIp0gfk4CXDD1GUC0PLBdTJCDJJg-E_ZrmmbdlF2kw8vJqw885nMDYxzxoC2UaUZNr0QQP8vIo68nRZEW-337AkqxgJNiaKggH4SMmYg/s400/IMG_3261.jpg" width="400" /></a></div>
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Notice how, because of the symmetry, it doesn't matter which side is facing up.</div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjLWAJvbysZf5LPVFIJqX8GXMGDFXXMuIxoDiA82oW2s-b8DvmCsvhV5ZjCYQPbXKCAHJCUIug5v4pzjwaWJ0uXfw_XOvYXf4WGhlPTk_Z56Qelo7zxu4RnVVOZWo9rDVKl6kgQX1mwSDs/s1600/IMG_3262.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="310" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjLWAJvbysZf5LPVFIJqX8GXMGDFXXMuIxoDiA82oW2s-b8DvmCsvhV5ZjCYQPbXKCAHJCUIug5v4pzjwaWJ0uXfw_XOvYXf4WGhlPTk_Z56Qelo7zxu4RnVVOZWo9rDVKl6kgQX1mwSDs/s320/IMG_3262.jpg" width="320" /></a></div>
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This pattern also has some neat rotational symmetry, but that's another story...</div>
Amyhttp://www.blogger.com/profile/14741075245934673435noreply@blogger.com0tag:blogger.com,1999:blog-7434427193865578406.post-48174860324370137082013-09-14T02:38:00.000-07:002013-10-08T18:46:20.619-07:00Twisted Tessellations I - Translation<div class="separator" style="clear: both; text-align: center;">
<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEg02bdI5UfZPE63tcTfqeCeSWDwN1IIVJQzj-cLavZLUXaxnp6GlMMEXz3Sap6bnnK2OYqhWZ50weSK_3CmpCF4Zq887S7BLr4Wyqk3eGDEcr0ATK4KvILuRjY8M1S7kvp_H_hQSt8Mlu4/s1600/IMG_3182.JPG" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="300" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEg02bdI5UfZPE63tcTfqeCeSWDwN1IIVJQzj-cLavZLUXaxnp6GlMMEXz3Sap6bnnK2OYqhWZ50weSK_3CmpCF4Zq887S7BLr4Wyqk3eGDEcr0ATK4KvILuRjY8M1S7kvp_H_hQSt8Mlu4/s400/IMG_3182.JPG" width="400" /></a></div>
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This delightful tessellation I found on an intricately decorated door has inspired a four-part series on drawing the various types of tessellations. The word "tessellation" comes from the Latin <i>tessella, </i>which is a small square tile of stone or glass used to make mosaics. Thus a tessellation is a tiling of a plane using geometric shapes with no overlaps or gaps. </div>
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<a href="http://britton.disted.camosun.bc.ca/escher/pegasus.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="320" src="http://britton.disted.camosun.bc.ca/escher/pegasus.jpg" width="320" /></a></div>
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Original artwork by M. C. Escher. <a href="http://britton.disted.camosun.bc.ca/jbescher3.htm">Image source.</a> One type of tessellation is called a <b>translation</b>. Here, the shape is simply translated, or slid, across the plane and drawn again. And now, a fairly easy way to create translation tessellations of your own. </div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiRiuQSY2AK556pRk_8u-NK-9g86Gh_vO06aa1uiAfGdVz0xyqw-DLW0PsCMd1lsajIVYdJES3IthVnTbT65LQT56cgiVYt1g22lW5T0UlQOWascRH_FgxQZZokaYr_rWuhyphenhyphenO7gKvGzZo0/s1600/IMG_3248.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="190" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiRiuQSY2AK556pRk_8u-NK-9g86Gh_vO06aa1uiAfGdVz0xyqw-DLW0PsCMd1lsajIVYdJES3IthVnTbT65LQT56cgiVYt1g22lW5T0UlQOWascRH_FgxQZZokaYr_rWuhyphenhyphenO7gKvGzZo0/s200/IMG_3248.jpg" width="200" /></a><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjb22WmICNN-f8JPLhBv_nfEg15tZPr4tF_id_oJKxRGtVDrLrLryhNOE6tontG4scyi6l1nthvt6XUbkpsiapGE22RbeMIoXIQmu3vVq6YdOcVxkwZ_VPmaUgImsh-5dkrRvGgb7SRQEs/s1600/IMG_3249.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="200" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjb22WmICNN-f8JPLhBv_nfEg15tZPr4tF_id_oJKxRGtVDrLrLryhNOE6tontG4scyi6l1nthvt6XUbkpsiapGE22RbeMIoXIQmu3vVq6YdOcVxkwZ_VPmaUgImsh-5dkrRvGgb7SRQEs/s200/IMG_3249.jpg" style="cursor: move;" width="188" /></a></div>
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<b>Step 1 </b>- Take a square piece of paper (this is a 2" x 2" square) and draw any sort of squiggle that you want. Thicker paper works better because we're going to be using it like a stencil later on; here I'm using colorful index cards. </div>
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<b>Step 2 </b>- Cut out the squiggly bit. </div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhapbaTka-H4RuIbFCDeTmDNYFa7-vY8o7rXXJ4TUmHFBWQdaFYDRUXEhLJdV33Af-A0hEwhJ2JWIAU4aC7HiDjXBp353PVK3gIiKFzkXuo5V6TESXFvdYsTWgVEd8YeXSeufYndz17vns/s1600/IMG_3250.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="200" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhapbaTka-H4RuIbFCDeTmDNYFa7-vY8o7rXXJ4TUmHFBWQdaFYDRUXEhLJdV33Af-A0hEwhJ2JWIAU4aC7HiDjXBp353PVK3gIiKFzkXuo5V6TESXFvdYsTWgVEd8YeXSeufYndz17vns/s200/IMG_3250.jpg" width="163" /></a> <a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhcsQNwsuUgoouM2oYJHzAQmoHcDp8_GNrOKDc1f4UrwQsM45PYc_YvXl3jEBPNTq_P6s5a8sMH5TtsXVCSET6S8Zm740MTGjxLbNV62cusZn4liTiyrWvy6iqhqdRS37yoPQqqcnoxYKM/s1600/IMG_3251.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="200" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhcsQNwsuUgoouM2oYJHzAQmoHcDp8_GNrOKDc1f4UrwQsM45PYc_YvXl3jEBPNTq_P6s5a8sMH5TtsXVCSET6S8Zm740MTGjxLbNV62cusZn4liTiyrWvy6iqhqdRS37yoPQqqcnoxYKM/s200/IMG_3251.jpg" width="170" /></a></div>
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<b style="text-align: left;">Step 3 </b><span style="text-align: left;">- Slide the squiggly bit down and tape it to the edge opposite the one you cut it from. </span><br />
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<b>Step 4 </b>- Pick one of the remaining two sides and draw another squiggle. </div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEi8Z8SYmTG5jiywLp2xtUuSMwFDlYoA8qeQnIY9jk2F6cr6rUneIUxpmfVAzP2waTW6F0xcxiJADTmvnD86_RC27PsCJV9RaYUTbwmcJHhI3rRm7HP8zs69BgX3kH7GahancixYwQKtyfM/s1600/IMG_3252.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="188" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEi8Z8SYmTG5jiywLp2xtUuSMwFDlYoA8qeQnIY9jk2F6cr6rUneIUxpmfVAzP2waTW6F0xcxiJADTmvnD86_RC27PsCJV9RaYUTbwmcJHhI3rRm7HP8zs69BgX3kH7GahancixYwQKtyfM/s200/IMG_3252.jpg" width="200" /></a><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiHDG-D-6kIGbHUEGmlfRuu_6pm4qy-r9jbjcWyMOgrpPy4O0DvSv3cjfABiEbSkktFQWNSdU2DAVELF09nq6uExmPTyT4kdXPyNMn3nKBP7q3IgxZbTn5Cxh7eVIsh7DrQHDhfyT7b5P4/s1600/IMG_3253.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="179" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiHDG-D-6kIGbHUEGmlfRuu_6pm4qy-r9jbjcWyMOgrpPy4O0DvSv3cjfABiEbSkktFQWNSdU2DAVELF09nq6uExmPTyT4kdXPyNMn3nKBP7q3IgxZbTn5Cxh7eVIsh7DrQHDhfyT7b5P4/s200/IMG_3253.jpg" width="200" /></a></div>
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<b>Step 5 </b>- Repeat Step 3 with the second squiggly bit. </div>
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<b>Step 6 </b>- Now your stencil is ready to use. Hold the shape still on a piece of paper and trace around it. </div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiTV7Is1STGAahtt3yjsY2WCb5YCdCXuzjkvUSvk8Am2MdjQXZOhR3Rlefi8VBK70vFpSz17in5i7oYDyFYxUHICHTf-oZvhdYGTzH9iXzMHLCej-Yk9cQb4cpPrti_IHRC8OBmHs-_Bxs/s1600/IMG_3254.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="400" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiTV7Is1STGAahtt3yjsY2WCb5YCdCXuzjkvUSvk8Am2MdjQXZOhR3Rlefi8VBK70vFpSz17in5i7oYDyFYxUHICHTf-oZvhdYGTzH9iXzMHLCej-Yk9cQb4cpPrti_IHRC8OBmHs-_Bxs/s400/IMG_3254.jpg" width="278" /></a></div>
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You can now continue the pattern by sliding the whole shape up or down/left or right and lining it up with the already drawn edge and tracing again. </div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiciXHslc0H286oWsG_hpc5clBAyEj7daCtMDsW1V_PDiHq9G17Cv1RkcJ3MsPA4d4H7pfrZg6S35wcZ1eAZNl1Lbw_u37-iwoX5HmHe4IySCjlwulBQoUIOHX96dDhHVjwNp-7DSz_QeQ/s1600/IMG_3255.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="368" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiciXHslc0H286oWsG_hpc5clBAyEj7daCtMDsW1V_PDiHq9G17Cv1RkcJ3MsPA4d4H7pfrZg6S35wcZ1eAZNl1Lbw_u37-iwoX5HmHe4IySCjlwulBQoUIOHX96dDhHVjwNp-7DSz_QeQ/s400/IMG_3255.jpg" width="400" /></a></div>
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This super simple method can be continued on forever and you can expand your tessellation indefinitely (or until you run out of paper, whichever comes first). </div>
Amyhttp://www.blogger.com/profile/14741075245934673435noreply@blogger.com0tag:blogger.com,1999:blog-7434427193865578406.post-74804830210210084222013-09-08T22:06:00.002-07:002013-11-07T19:32:45.898-08:00Visualizing Solids of Revolution <div style="text-align: center;">
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A project two of my friends and I did for our Multivariable class, showing different visualizations, examples, and applications of finding the volume of a solid of revolution using disks, washers, and shells - a concept we learned last year. </div>
Amyhttp://www.blogger.com/profile/14741075245934673435noreply@blogger.com0tag:blogger.com,1999:blog-7434427193865578406.post-57993665249210091412013-09-02T20:39:00.002-07:002013-09-02T20:39:43.321-07:00Fibonnaci Pinecone<div class="separator" style="clear: both; text-align: center;">
<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEh3aPT9D_DJ_-BLdGZGMhjOVRn2RWVAb0CrrS63qCvPKqtYVU1XPv9UZLn2IudrL-7STExbw9E5ORbOTkcC35lOU2prktRWJfhi0X-pLR433vBPD0KJJXVuhBaUgItKlQ7tDXKyHYH5H_0/s1600/IMG_2057.JPG" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="300" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEh3aPT9D_DJ_-BLdGZGMhjOVRn2RWVAb0CrrS63qCvPKqtYVU1XPv9UZLn2IudrL-7STExbw9E5ORbOTkcC35lOU2prktRWJfhi0X-pLR433vBPD0KJJXVuhBaUgItKlQ7tDXKyHYH5H_0/s400/IMG_2057.JPG" width="400" /></a></div>
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So I really like pinecones - I have a bunch just lying around my room because I find them very spirally and pretty. But in addition to that, they're beautifully mathematical. All you have to do is count the number of spirals; this pinecone has 8 going this way... </div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEi_t0hhu8zTY8IV3oDFM-le2cevptt4aGHPXWgsTjlJ3_ZGmwp_ieCzTMn121uypiGQo06kciV8F2QNbbW1pSHJg29ougNlijfgbz9on9jiRV_TOoMgcEEFmPl6Tf4UBz7P54EKPCvuhko/s1600/IMG_2058.JPG" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="300" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEi_t0hhu8zTY8IV3oDFM-le2cevptt4aGHPXWgsTjlJ3_ZGmwp_ieCzTMn121uypiGQo06kciV8F2QNbbW1pSHJg29ougNlijfgbz9on9jiRV_TOoMgcEEFmPl6Tf4UBz7P54EKPCvuhko/s400/IMG_2058.JPG" width="400" /></a></div>
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And 13 the other way. Two of my other pinecones have 3 spirals and 5 spirals, and 5 and 8 spirals. Sound familiar? These are adjacent Fibonacci numbers - seriously, the series you get by adding the two previous numbers that goes 1, 1, 2, 3, 5, 8, 13, 21, 34... shows up everywhere in nature, from sunflower seeds to artichokes to pineapples. This is because these Fibonacci arrangements result in the optimal packing of seeds/leaves/plant bits, so that they are all uniformly spaced and can maximize the amount of sunlight they receive. The more you look for these instances of Fibonacci numbers in the world around you, the more you'll find them - delightful glimpses into the mathematics in which our universe is written. </div>
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<br />Amyhttp://www.blogger.com/profile/14741075245934673435noreply@blogger.com0tag:blogger.com,1999:blog-7434427193865578406.post-35969164010690542292013-08-29T20:20:00.002-07:002013-08-29T20:20:33.618-07:00From the Doodle Book III<div class="separator" style="clear: both; text-align: center;">
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<span style="font-family: Georgia, Times New Roman, serif; font-size: large;"><span style="background-color: white; line-height: 20px; text-align: left;"><b>"One of the most amazing things about mathematics is the people who do math aren't usually interested in application, because mathematics itself is truly a beautiful art form. It's structures and patterns, and that's what we love, and that's what we get off on." </b></span></span><span style="font-family: Georgia, 'Times New Roman', serif; font-size: large; line-height: 20px; text-align: left;">- Danica McKellar </span></div>
Amyhttp://www.blogger.com/profile/14741075245934673435noreply@blogger.com0tag:blogger.com,1999:blog-7434427193865578406.post-45353905649789703312013-08-26T23:37:00.000-07:002013-08-26T23:37:12.359-07:00Another Proof Without Words<div class="separator" style="clear: both; text-align: center;">
<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiGsxh6p-AW9tMZA8tluCPXW4_WaH9woVh87joS8cTUSqhCfZE-V4HExexe8rAOFzQZIq2sobwOovT9BFYJg3PQDvmCy_LSjg2vrr62_756uuBvu7yczp6aVG_wkAxpWjBsj37Wh_eRVR4/s1600/IMG_1898.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="388" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiGsxh6p-AW9tMZA8tluCPXW4_WaH9woVh87joS8cTUSqhCfZE-V4HExexe8rAOFzQZIq2sobwOovT9BFYJg3PQDvmCy_LSjg2vrr62_756uuBvu7yczp6aVG_wkAxpWjBsj37Wh_eRVR4/s400/IMG_1898.jpg" width="400" /></a></div>
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<span style="font-family: inherit;">Here's another pretty famous <a href="http://coffeecupcardioids.blogspot.com/2013/08/proof-without-words.html">proof without words</a> - it shows that the sum of the first n odd integers is <span style="background-color: white; color: #131313; line-height: 20px; text-align: left;">n</span><sup style="background-color: white; color: #131313; line-height: 20px; text-align: left;">2</sup>. Starting with the upper left hand corner, we have 1 red bead. Then adding 3 clear beads (3 being the 2nd odd integer), we get <span style="background-color: white; color: #131313; line-height: 20px; text-align: left;">2</span><sup style="background-color: white; color: #131313; line-height: 20px; text-align: left;">2</sup> which is 4 beads in total. Adding the 3rd odd integer 5, we get <span style="background-color: white; color: #131313; line-height: 20px; text-align: left;">3</span><sup style="background-color: white; color: #131313; line-height: 20px; text-align: left;">2</sup> = 9 beads. </span></div>
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<span style="font-family: inherit;">Said another way, </span></div>
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<span style="font-family: inherit;"><span style="background-color: white; color: #131313; line-height: 20px;">1=1=1</span><sup style="background-color: white; color: #131313; line-height: 20px;">2</sup></span></div>
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<span style="font-family: inherit;"><span style="background-color: white; color: #131313; line-height: 20px;">1+3=4=2</span><sup style="background-color: white; color: #131313; line-height: 20px;">2</sup></span><br />
<span style="font-family: inherit;"><span style="background-color: white; color: #131313; line-height: 20px;">1+3+5=9=3</span><sup style="background-color: white; color: #131313; line-height: 20px;">2</sup></span><br />
<span style="font-family: inherit;"><span style="background-color: white; color: #131313; line-height: 20px;">1+3+5+7=16=4</span><sup style="background-color: white; color: #131313; line-height: 20px;">2</sup></span><br />
<span style="font-family: inherit;"><span style="background-color: white; color: #131313; line-height: 20px;">1+3+5+7+9=25=5</span><sup style="background-color: white; color: #131313; line-height: 20px;">2</sup></span><br />
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You could realize this via algebra as well. Consider the two consecutive square numbers <span style="background-color: white; color: #131313; line-height: 20px;">n</span><sup style="background-color: white; color: #131313; line-height: 20px;">2</sup> and (<span style="background-color: white; color: #131313; line-height: 20px;">n+1)</span><sup style="background-color: white; color: #131313; line-height: 20px;">2</sup>. Expanding <span style="text-align: left;">(</span><span style="background-color: white; color: #131313; line-height: 20px; text-align: left;">n+1)</span><sup style="background-color: white; color: #131313; line-height: 20px; text-align: left;">2</sup>, you get <span style="background-color: white; color: #131313; line-height: 20px;">n</span><sup style="background-color: white; color: #131313; line-height: 20px;">2</sup>+2n+1. Then the difference between one square number <span style="background-color: white; color: #131313; line-height: 20px;">n</span><sup style="background-color: white; color: #131313; line-height: 20px;">2</sup> and the next square number <span style="text-align: left;">(</span><span style="background-color: white; color: #131313; line-height: 20px; text-align: left;">n+1)</span><sup style="background-color: white; color: #131313; line-height: 20px; text-align: left;">2</sup> is simply (<span style="background-color: white; color: #131313; line-height: 20px;">n</span><sup style="background-color: white; color: #131313; line-height: 20px;">2</sup>+2n+1)-<span style="background-color: white; color: #131313; line-height: 20px;">n</span><sup style="background-color: white; color: #131313; line-height: 20px;">2</sup> = 2n+1. So to get from the nth square number to the next square number, you just add the next odd number, 2n+1.Amyhttp://www.blogger.com/profile/14741075245934673435noreply@blogger.com0tag:blogger.com,1999:blog-7434427193865578406.post-56465818681337238652013-08-25T20:11:00.002-07:002013-08-25T20:15:19.964-07:00Apples and Solids of Revolution<span style="text-align: center;">Here's an idea of how to visualize calculating the volume of solids of revolution with disks. You'll need a sliceable object of your choice - I chose an apple - as well as a ruler and a knife. The goal will be to cut up the apple into increasingly more slices and show that the more disks you have, the more accurate your volume estimate is. </span>
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Before slicing up the apple, I found its volume via displacement to be 121 cubic centimeters. It was (about) 5 cm high, so I marked off 1 cm slices. </div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjazWeybQUY37IdaWPsikUgEZFhXw5OljRxfVROOnBwlxK0nZOlH1fnLDsEryIeSaeIeCWpzV9bI5qgwB_svkDl6qGdfk0WL7VfXWnRx555VF5IsyZAoEqvtBiBiU294SMtBeHCmxXaI4g/s1600/IMG_3105+2.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="300" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjazWeybQUY37IdaWPsikUgEZFhXw5OljRxfVROOnBwlxK0nZOlH1fnLDsEryIeSaeIeCWpzV9bI5qgwB_svkDl6qGdfk0WL7VfXWnRx555VF5IsyZAoEqvtBiBiU294SMtBeHCmxXaI4g/s400/IMG_3105+2.jpg" width="400" /></a></div>
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Then I cut the apple in half, measured the radius, and calculated the volume as if the apple were a single cylinder/disk with a height of 5 cm. Obviously this is going to be a gross overestimation of the volume, as the apple is clearly not a cylinder, but we can improve our estimate by slicing again. </div>
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With two cuts, you now have two different radii. Unfortunately, the disks have different heights, though this could be fixed if you instead opted for a slicing method in which you take each previous slice and cut it in half (resulting in exponentially more slices). </div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjA6cmJdVJ2zqOYN4SThRlWagURKOH85ptmyNE7-kVKVa-fdqBLWndyau7GIlyupqDMFWIsOtIBkw4a-mHM60R3S9YjSBJbNp7u32Cljfmgee9imHVxZJN6vOyZZhXS0TNmvYSSkutleGU/s1600/IMG_3107.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="300" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjA6cmJdVJ2zqOYN4SThRlWagURKOH85ptmyNE7-kVKVa-fdqBLWndyau7GIlyupqDMFWIsOtIBkw4a-mHM60R3S9YjSBJbNp7u32Cljfmgee9imHVxZJN6vOyZZhXS0TNmvYSSkutleGU/s400/IMG_3107.jpg" width="400" /></a></div>
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Conceptual drawing of an apple as approximated by cylindrical disks</div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiWDYMGjh1HNDwxqSm-V7L99M9PMTcRIKAEQu1XWvg6Et3J2GDFGXqqARA1jNonXGA_ZG32r_FYuAg3vqVbXifIp3pd1nEWSwcdBHRrP7nVc_2dQPTW96NmUWlWeV1NIPsKW3uIZuodEOk/s1600/IMG_3114.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="371" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiWDYMGjh1HNDwxqSm-V7L99M9PMTcRIKAEQu1XWvg6Et3J2GDFGXqqARA1jNonXGA_ZG32r_FYuAg3vqVbXifIp3pd1nEWSwcdBHRrP7nVc_2dQPTW96NmUWlWeV1NIPsKW3uIZuodEOk/s400/IMG_3114.jpg" width="400" /></a></div>
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In the end, the approximate volumes I got were as follows: </div>
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1 cut (2 disks) - 171.05972 <span style="background-color: white; color: #131313; line-height: 20px;">cm</span><sup style="background-color: white; color: #131313; line-height: 20px;">3</sup> </div>
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2 cuts (3 disks) - 165.95463 <span style="background-color: white; color: #131313; line-height: 20px;">cm</span><sup style="background-color: white; color: #131313; line-height: 20px;">3</sup></div>
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3 cuts (4 disks) - 157.76031 <span style="background-color: white; color: #131313; line-height: 20px;">cm</span><sup style="background-color: white; color: #131313; line-height: 20px;">3</sup></div>
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4 cuts (5 disks) - 126.77023 <span style="background-color: white; color: #131313; line-height: 20px;">cm</span><sup style="background-color: white; color: #131313; line-height: 20px;">3</sup></div>
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The approximations nicely approach the true volume of 121 <span style="background-color: white; color: #131313; line-height: 20px;">cm</span><sup style="background-color: white; color: #131313; line-height: 20px;">3</sup>. Unfortunately, it becomes quite difficult to cut thinner cylinders accurately (unless you happen to be a master fruit chopper ninja). You could instead opt to continue your apple slicing on the computer - simply take a picture and measure the slice radii virtually. This particular image shows slices every quarter centimeter. </div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEj9c4n_wgckz__AYmWxuIG9pUx2hw6No2_xsc6BH8m1yewwMLU1YWbBsrh9e-mXYudwGC_wr5vpRh9pH91BoylwJay0AD-wFPEjtqZ8D9VJZ3LU9BCY6SW_8Yb_l-wZFQtvn7i7Che4Fco/s1600/Screen+Shot+2013-08-25+at+7.07.40+PM.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="353" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEj9c4n_wgckz__AYmWxuIG9pUx2hw6No2_xsc6BH8m1yewwMLU1YWbBsrh9e-mXYudwGC_wr5vpRh9pH91BoylwJay0AD-wFPEjtqZ8D9VJZ3LU9BCY6SW_8Yb_l-wZFQtvn7i7Che4Fco/s400/Screen+Shot+2013-08-25+at+7.07.40+PM.png" width="400" /></a></div>
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The idea you're illustrating here is that as you use more and more disks, the approximation approaches the true volume. If you were to cut the apple into infinitely many (infinitesimally thin) slices and add them up, you'd get the exact volume, which is what integrals allow you to do. </div>
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Alternatively, you could take this apple slicing even further and plot some points along the edge of the apple, solve for the equation of the polynomial going through all of those points, and then do the actual integral. I however, am going to go eat these apple slices. </div>
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Amyhttp://www.blogger.com/profile/14741075245934673435noreply@blogger.com0tag:blogger.com,1999:blog-7434427193865578406.post-58966109114757687542013-08-25T14:18:00.000-07:002013-08-25T19:13:11.786-07:00A Little Bit of Binary <div class="separator" style="clear: both; text-align: center;">
<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgsJ2Nh4ci5ly5zn2JLBgnx5mJalqNDYczr3J3R3uDQbE_ZvhjjQx9l0jbMURij2Y-415tnYc-Byh4Tu9sza61bdDOYCbX9R0Fm8BaxU3RRcHrq8c2CufI4qOx-stKTkN2gBW2Ogy87AmA/s1600/IMG_1897.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="640" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgsJ2Nh4ci5ly5zn2JLBgnx5mJalqNDYczr3J3R3uDQbE_ZvhjjQx9l0jbMURij2Y-415tnYc-Byh4Tu9sza61bdDOYCbX9R0Fm8BaxU3RRcHrq8c2CufI4qOx-stKTkN2gBW2Ogy87AmA/s640/IMG_1897.jpg" width="248" /></a></div>
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Saw some flat glass beads lying around my house and figured they had some potential for mathematical craftiness - here are the numbers 0 through 8 in binary (taking the clear beads to be 0's and the red ones to be 1's). I would've liked to go higher but I ran out of red beads, so I switched to graph paper. </div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEh0nvIFc6V5GDdzZa1GyEJFblzl7di_NYRhsk_5F9vPjFxislSeLdKCAfNI3fUnQMcQVbdVPeFaiykI1OGx3XGZx6uapn7l6EnRMstyEBK1ebFQcwMpbkkoM9MKuVu9oUNF-38FtS341hY/s1600/IMG_1899.JPG" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="400" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEh0nvIFc6V5GDdzZa1GyEJFblzl7di_NYRhsk_5F9vPjFxislSeLdKCAfNI3fUnQMcQVbdVPeFaiykI1OGx3XGZx6uapn7l6EnRMstyEBK1ebFQcwMpbkkoM9MKuVu9oUNF-38FtS341hY/s400/IMG_1899.JPG" width="92" /></a><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiNb7WcUmEx1MsZnDv4RMOYSo5po9185E129CKRLoNXvn-9fyiDX82j_M0hs5U12k9-WRFsx3I0Q1RXhbzn2gKCXJqUedGoI2uehVv6otiLgF-KN9V98880kl2u8I5QJ0v90u64KYJ09Kw/s1600/IMG_1901.JPG" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="400" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiNb7WcUmEx1MsZnDv4RMOYSo5po9185E129CKRLoNXvn-9fyiDX82j_M0hs5U12k9-WRFsx3I0Q1RXhbzn2gKCXJqUedGoI2uehVv6otiLgF-KN9V98880kl2u8I5QJ0v90u64KYJ09Kw/s400/IMG_1901.JPG" width="123" /></a></div>
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Check it out! I marked off blue lines between powers of 2 and one less than power of 2 (0 and 1, 1 and 2, 3 and 4, etc.) and there's this really neat pattern: if you folded along any of those lines, the boxes on the other side are the exact opposite. Say you folded on the line between 15 and 16, you could take all the numbers you've already drawn from 0 to 15, reflect them over the line, and invert the colors to get the next 16 numbers. This suggests an easy way to quickly color in binary. You'll need some graph paper and a pen that bleeds through (Sharpies work well) </div>
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<b>Step 1: </b>Start coloring and stop when you're one less than a power of 2 (Technically you could start with just 0 and 1 and build up from there). I've done from 0 to 7. Take your paper, fold it along the blue line, and flip it over so that the back of the paper is facing you. Make sure you can still see the squares through the paper though; here I've taped my paper to a window. I also counted down 8 more squares and marked this so that I know where to stop. </div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhlUonadBzYkOSwVyx83XS4APLmtLxLMDWFlD2t4eIANomRKk52q_2io4mqhOKzoNSvIfGYQ8l60sz-MJeNuPyj9LJFeJdRMSpc4dtFlAKQxHQw98cXDo8C1HSv48BbbDH8ytxjxvY31OE/s1600/IMG_1904.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="200" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhlUonadBzYkOSwVyx83XS4APLmtLxLMDWFlD2t4eIANomRKk52q_2io4mqhOKzoNSvIfGYQ8l60sz-MJeNuPyj9LJFeJdRMSpc4dtFlAKQxHQw98cXDo8C1HSv48BbbDH8ytxjxvY31OE/s200/IMG_1904.jpg" width="191" /></a><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEi9bSS4C_MdJwMciXn1__8Q59oD5P_DmzUmfts0FT63fDBTMOpP0IkETtwQhyphenhyphenwjJ-a0ysxZxn6d0dcuyrqRo90LgUlC346fVIqI7dwrluD_WbWInw9hnsxYJrH-rR0DMowZFPXRal7Y-bI/s1600/IMG_1906.JPG" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="200" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEi9bSS4C_MdJwMciXn1__8Q59oD5P_DmzUmfts0FT63fDBTMOpP0IkETtwQhyphenhyphenwjJ-a0ysxZxn6d0dcuyrqRo90LgUlC346fVIqI7dwrluD_WbWInw9hnsxYJrH-rR0DMowZFPXRal7Y-bI/s200/IMG_1906.JPG" width="182" /></a></div>
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<b>Step 2: </b>Color in all the squares that you didn't already color on the other side. When you're done, it'll look like a solid red block of squares. </div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEg3NHRDSSov1dA5ii30QBAn6jNQ3_RNH-WW7sLgxPJm8QcuN8wUdKY3vGsla0OtX-69E2gBEihbBRkmekqsEnGeroOtTQVZBKMVDt-n_-VIGru5ualkEod7jpUPglsF2R2ta03xcj367PU/s1600/IMG_1909.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="320" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEg3NHRDSSov1dA5ii30QBAn6jNQ3_RNH-WW7sLgxPJm8QcuN8wUdKY3vGsla0OtX-69E2gBEihbBRkmekqsEnGeroOtTQVZBKMVDt-n_-VIGru5ualkEod7jpUPglsF2R2ta03xcj367PU/s320/IMG_1909.jpg" width="300" /></a></div>
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<b>Step 3: </b>Unfold and voila! Now you have twice as much binary coloring - from 0 to 15. You could then take this and repeat the steps again with the new line to get all the way up to 31 and go on generating exponentially more binary (that is, until you run out of graph paper).<br />
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhYKWVxS0Fcs_3XtUP_v7helvvqMw-0YdoeLBuX_6nsC69bnVZa-vqKbSIXLHwcl-5PboOuiNylBzTKrXJpE58HKvTIwaeIaGq14Obf76rtRntvBlRC-RP9qqb6iOzLf3mIjGqzI_a9GFE/s1600/IMG_1910.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="320" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhYKWVxS0Fcs_3XtUP_v7helvvqMw-0YdoeLBuX_6nsC69bnVZa-vqKbSIXLHwcl-5PboOuiNylBzTKrXJpE58HKvTIwaeIaGq14Obf76rtRntvBlRC-RP9qqb6iOzLf3mIjGqzI_a9GFE/s320/IMG_1910.jpg" width="240" /></a></div>
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Amyhttp://www.blogger.com/profile/14741075245934673435noreply@blogger.com0tag:blogger.com,1999:blog-7434427193865578406.post-79914270615454441692013-08-22T20:41:00.000-07:002013-08-22T20:41:00.563-07:00Cuboctahedron from Magnets<div class="separator" style="clear: both; text-align: center;">
<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEip97axyQBuotIgj1VKfDU1yvEwh5BtOAQ-im3UBVvOV5_ev3yJDxmlDt66bNFohOTH_LW8NhxGxwQ8eKdY8KRI4xVkxYf1jK_8YgR5XmkRoziyVktmv-XKLMnmzGhxz-18TZ7FydXXxwU/s1600/IMG_1892.JPG" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="400" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEip97axyQBuotIgj1VKfDU1yvEwh5BtOAQ-im3UBVvOV5_ev3yJDxmlDt66bNFohOTH_LW8NhxGxwQ8eKdY8KRI4xVkxYf1jK_8YgR5XmkRoziyVktmv-XKLMnmzGhxz-18TZ7FydXXxwU/s400/IMG_1892.JPG" width="396" /></a></div>
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I made something today. After some internet searchings, I discovered that this is called a cuboctahedron. </div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiDIDwgP2wnx29cTR1okKYOlziK8cxb7kge0NfWWfWK7D6fTJburGOerV8tYJvjy_LiUNjRlUsXpCZpZUcJmV9GTyMqhGTRdOPbG_4srbT_A3S6sJqMSaGGe9FC83OfTI6LMM1hBfkmOCc/s1600/IMG_1893.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="400" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiDIDwgP2wnx29cTR1okKYOlziK8cxb7kge0NfWWfWK7D6fTJburGOerV8tYJvjy_LiUNjRlUsXpCZpZUcJmV9GTyMqhGTRdOPbG_4srbT_A3S6sJqMSaGGe9FC83OfTI6LMM1hBfkmOCc/s400/IMG_1893.jpg" width="393" /></a></div>
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It is quite pretty and symmetric. The red pieces show that it can be made of pyramids and tetrahedrons.</div>
Amyhttp://www.blogger.com/profile/14741075245934673435noreply@blogger.com0tag:blogger.com,1999:blog-7434427193865578406.post-3331379998396017322013-08-20T17:24:00.000-07:002013-08-25T14:21:55.705-07:00Proof Without Words<div class="separator" style="clear: both; text-align: center;">
<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgezz7xA5Z-0LGBFmi4XtomzBlhm8E3IdmJwKHwhYkbCVpFRWJmtQuJ2N8T_eBPXUZuN6gykI8JDtil0Uwq61OSmbdzHV1CImBm_xrOVqOSLX1Ek3bomEvZwZmwVZsmCFZxD7Bl1gZOxP4/s1600/IMG_1888.JPG" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="400" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgezz7xA5Z-0LGBFmi4XtomzBlhm8E3IdmJwKHwhYkbCVpFRWJmtQuJ2N8T_eBPXUZuN6gykI8JDtil0Uwq61OSmbdzHV1CImBm_xrOVqOSLX1Ek3bomEvZwZmwVZsmCFZxD7Bl1gZOxP4/s400/IMG_1888.JPG" width="387" /></a></div>
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I love this. A proof without words is, as the name suggests, an image that conveys something without requiring any explanatory text. This one, which Archimedes came up with, shows that 1/4 + 1/16 + 1/64 + 1/256... = 1/3. Seeing the entire square as equalling 1, the green (I can call that green, yeah? It's more of a chartreuse I suppose, but I digress...) squares color in 1/4 of the whole square, then 1/4 of 1/4 ((1/4)^2 = 1/16), then 1/4 of 1/4 of 1/4 ((1/4)^3 = 1/64) etc. and you can see that each green square is one of three of equal size. So in total the square is divided into three equal sets of squares, one of which is colored green, thus the green squares take up 1/3 of the whole square. While perhaps not as rigorous as a traditional proof, it is so very elegant and beautiful and it makes you wonder, can you prove other geometric series this way? </div>
<br />Amyhttp://www.blogger.com/profile/14741075245934673435noreply@blogger.com0tag:blogger.com,1999:blog-7434427193865578406.post-85103002910497871912013-08-17T22:26:00.001-07:002013-08-17T22:26:51.194-07:00A Study in Orange<div class="separator" style="clear: both; text-align: center;">
<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhGs3IDQBfhn3op-dui7npwY1IZjiLUI-AyWV4geXR9W2W5eYv0BPIqi-MticdcvbA2GoRZyae9JbnqVzF0CpcbXlm0t1GxUgt2fuE2L1PqHUXq51rlrgTlQ5eVlrOByb4CwR5Pfj9ELEk/s1600/IMG_1882.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="380" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhGs3IDQBfhn3op-dui7npwY1IZjiLUI-AyWV4geXR9W2W5eYv0BPIqi-MticdcvbA2GoRZyae9JbnqVzF0CpcbXlm0t1GxUgt2fuE2L1PqHUXq51rlrgTlQ5eVlrOByb4CwR5Pfj9ELEk/s400/IMG_1882.jpg" width="400" /></a></div>
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An (approximately) spherical orange with the six planes of tetrahedral symmetry, done with rubber bands. Who knows, maybe one day I'll be brave enough to try icosahedral/dodecahedral symmetry (fifteen planes). Now if only I could manage to cut this orange along these planes...</div>
<br />Amyhttp://www.blogger.com/profile/14741075245934673435noreply@blogger.com0tag:blogger.com,1999:blog-7434427193865578406.post-86064178424482995572013-08-16T13:47:00.002-07:002013-08-16T13:47:50.245-07:00Upcycled Bottle Top Ring Construction<div class="separator" style="clear: both; text-align: center;">
<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhMsGNltLjSfo0Y-YFSxK0PIgwqdH-ZHy0g4vhqiMkt9rCtQ9tdzazjkZ_ETCPshfMt8yKp03h9YFULCSw0VFTNooBQ9H0PmQQSQimjLA8DRZFAMA8wJ4LC9XA2GyHn3NYcj_gGqAJd-Fk/s1600/IMG_1873.JPG" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="400" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhMsGNltLjSfo0Y-YFSxK0PIgwqdH-ZHy0g4vhqiMkt9rCtQ9tdzazjkZ_ETCPshfMt8yKp03h9YFULCSw0VFTNooBQ9H0PmQQSQimjLA8DRZFAMA8wJ4LC9XA2GyHn3NYcj_gGqAJd-Fk/s400/IMG_1873.JPG" width="358" /></a></div>
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I had several of those rings from the tops of plastic bottles, so I made a fun little thing with them. The rings are secured to one another with pieces of duct tape. The original intention was to make an octahedron since I had eight rings (faces), but it kind of turned into something else...at the top and bottom, four faces come together to meet at one vertex, but around the middle, instead of having four points where four faces meet up, there are eight points where three faces come together.<br />
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiaVYNTQ91Ss67uKU83RnRAdi66MWC0WJtGcWQRBI9VgpLM1KLbZp7lL0Z3Nnlg9UxSB_XCpvegon1bHOpFyTuI6S3apZ0Sckw_JkanOzWO41NXEbEiKNX0H9ZYxcACZFDW5z3uRh6uM98/s1600/IMG_1875.JPG" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="333" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiaVYNTQ91Ss67uKU83RnRAdi66MWC0WJtGcWQRBI9VgpLM1KLbZp7lL0Z3Nnlg9UxSB_XCpvegon1bHOpFyTuI6S3apZ0Sckw_JkanOzWO41NXEbEiKNX0H9ZYxcACZFDW5z3uRh6uM98/s400/IMG_1875.JPG" width="400" /></a></div>
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Here's a pic of one of the eight vertexes where three faces meet. It's quite fun to play with and it actually bounces a little bit. I might dismantle it when I've collected another four rings and repurpose it into a dodecahedron though.<br />
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhJOKtxWy8Ar2kQupV4t5u1hwVnWjgroxLjHb47ETtYMTpQS3fKAQRyWGwEKKTabBr5GrM1uvdWc1YURxPSgZXkpSZKva9oIYxqyDCVueOc3KHGCATH-tkQhZqaWagpQ0MFoHZgNTt7nWI/s1600/IMG_1878.JPG" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="386" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhJOKtxWy8Ar2kQupV4t5u1hwVnWjgroxLjHb47ETtYMTpQS3fKAQRyWGwEKKTabBr5GrM1uvdWc1YURxPSgZXkpSZKva9oIYxqyDCVueOc3KHGCATH-tkQhZqaWagpQ0MFoHZgNTt7nWI/s400/IMG_1878.JPG" width="400" /></a></div>
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A shot from the top.</div>
Amyhttp://www.blogger.com/profile/14741075245934673435noreply@blogger.com0tag:blogger.com,1999:blog-7434427193865578406.post-42534103690348521412013-08-16T00:09:00.000-07:002013-08-16T00:09:06.978-07:00Optimization Problem via GeometrySo I was browsing my old calculus book (don't judge...yes, I browse my calculus book), and I saw this optimization problem -<br />
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEi97gy8vZSKMV12DjpKyK2rf1J3bLTV0mBK-5jadQSivQdf5YbNh7RXHU1RCUPPJ61QURdkbBPspMiZRXA5y6jqHKPho-LYyzQYpAGIzTs4GnTCVBCZWh2AZ1u5PIxL1IPW2hqNWmRlgUg/s1600/IMG_1868.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="276" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEi97gy8vZSKMV12DjpKyK2rf1J3bLTV0mBK-5jadQSivQdf5YbNh7RXHU1RCUPPJ61QURdkbBPspMiZRXA5y6jqHKPho-LYyzQYpAGIzTs4GnTCVBCZWh2AZ1u5PIxL1IPW2hqNWmRlgUg/s320/IMG_1868.jpg" width="320" /></a></div>
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<span style="font-family: Times, Times New Roman, serif;"><b>Two vertical poles PQ and ST are secured by a rope PRS going from the top of the first pole to a point R on the ground between the poles and then to the top of the second pole as in the figure. Show that the shortest length of such a rope occurs when <span style="background-color: white; line-height: 19px;">θ1=</span><span style="background-color: white; line-height: 19px;">θ2.</span></b></span></div>
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<span style="text-align: -webkit-auto;">I'm pretty sure I even did this question, but now looking back on it the answer seems obvious. </span></div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEj3dCt7bdMkZEE5GNkUsvZkPGQM5wHhxQmjQPfpKOLIBFlM0PQNLRvHpyTR1XBCS1imcEBaXZ9rwdT4AFLiHxji0UIYgx83R2_YvJ5tTfWPDhDSEHj1cUriHcKC6o3uJGHCTATKiV_8jLk/s1600/IMG_1869.JPG" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="320" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEj3dCt7bdMkZEE5GNkUsvZkPGQM5wHhxQmjQPfpKOLIBFlM0PQNLRvHpyTR1XBCS1imcEBaXZ9rwdT4AFLiHxji0UIYgx83R2_YvJ5tTfWPDhDSEHj1cUriHcKC6o3uJGHCTATKiV_8jLk/s320/IMG_1869.JPG" width="262" /></a></div>
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Because if you simply take either triangle PQR or RST and reflect it across line QT (the ground), now when you're looking for the shortest length of rope between points P and S, you know from geometry that this is just a straight line. Then you don't even need to do any calculus because <span style="font-family: Times, 'Times New Roman', serif; text-align: center;"><span style="background-color: white; line-height: 19px;">θ1</span></span> and <span style="font-family: Times, 'Times New Roman', serif; text-align: center;"><span style="background-color: white; line-height: 19px;">θ2</span></span> are vertical angles so they're congruent. </div>
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What bothered me a lot about most of my elementary and middle school math was that you would be taught how to do a certain kind of problem and then expected to follow that method exactly - there was never room for exploration or creativity or anything and an answer was somehow "wrong" if we didn't do it "the book's way." Personally, I think both the calculus and the geometry paths are equally valid and satisfying, just two different ways of telling the same story. </div>
<br />Amyhttp://www.blogger.com/profile/14741075245934673435noreply@blogger.com0tag:blogger.com,1999:blog-7434427193865578406.post-67731259039884521922013-08-15T00:17:00.000-07:002013-08-15T23:40:17.919-07:00From the Doodle Book II <div class="separator" style="clear: both; text-align: center;">
<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgmryL-N1M7UsZBASVbRgCbTlXyAInvbXPdZM8KEOAvEG2KG-b2eIda5unt-zIXzssGU0ejpUapEr2l3Q5zNFzoz1YADSCgVni3cB9KEQMQTDbL7jMHub70fEazugp1h9b95p5BOpC1Sdc/s1600/IMG_1859.JPG" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="370" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgmryL-N1M7UsZBASVbRgCbTlXyAInvbXPdZM8KEOAvEG2KG-b2eIda5unt-zIXzssGU0ejpUapEr2l3Q5zNFzoz1YADSCgVni3cB9KEQMQTDbL7jMHub70fEazugp1h9b95p5BOpC1Sdc/s400/IMG_1859.JPG" width="400" /></a></div>
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Done with Sharpies. A play on the idea of drawing an astroid as an envelope of lines connecting points on a square - I liked how these quarter astroids overlap and it makes for a pretty pattern.</div>
Amyhttp://www.blogger.com/profile/14741075245934673435noreply@blogger.com0tag:blogger.com,1999:blog-7434427193865578406.post-79393009772773870582013-08-14T01:43:00.001-07:002013-08-14T01:43:42.575-07:00Kabob Skewer Hyperboloid<div class="separator" style="clear: both; text-align: center;">
<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjTFhCbusC5xQXONwn9x54_KfHRZjYL5B2-qOF2kAe_vLyyLpKJL6jzVdfUt8YsR2J5jh1LA5APyqQiL3ZK6cVPvsQYDvRHavVLNOM9Ubr7P6iF2XpuEGyrJSWhgeFBZ9ZX8QT5IlxqaVA/s1600/IMG_1861.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="400" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjTFhCbusC5xQXONwn9x54_KfHRZjYL5B2-qOF2kAe_vLyyLpKJL6jzVdfUt8YsR2J5jh1LA5APyqQiL3ZK6cVPvsQYDvRHavVLNOM9Ubr7P6iF2XpuEGyrJSWhgeFBZ9ZX8QT5IlxqaVA/s400/IMG_1861.jpg" width="307" /></a></div>
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Here's a hyperboloid I made out of kabob skewers and rubber bands. It is largely reminiscent of this awesome double napped cone made out of string one of my math teachers had in his room. Anyhow, the hyperboloid is the surface obtained by revolving a hyperbola about its semi minor axis. It sits nestled around a jar here so that I could take a picture of it, but it's an awful lot of fun to play with - stretching/pushing it flatter and watching it change, giving the whole thing more curve by moving the rubber bands closer together, etc. It also makes a great hat. I like this construction because it illustrates the doubly ruled nature of the hyperboloid - that through every point, there are two distinct lines that pass through it on the surface. </div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEielgz6bNWGdawljYVLaf1eajLVCvnwPpH2k4ux5_KQRZSfHko0qdtl1DxF62f3g0D4kGPeLtTwBM6zHtt5N4tV6Fcb04uTuznS9ZDX-gWhjcfLxrzCH5oAOcUZ_WQ4RJltSQgpDeNJ7hE/s1600/IMG_1862.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="400" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEielgz6bNWGdawljYVLaf1eajLVCvnwPpH2k4ux5_KQRZSfHko0qdtl1DxF62f3g0D4kGPeLtTwBM6zHtt5N4tV6Fcb04uTuznS9ZDX-gWhjcfLxrzCH5oAOcUZ_WQ4RJltSQgpDeNJ7hE/s400/IMG_1862.jpg" width="300" /></a></div>
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An alternate view</div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjCC-Rt8XPrvBF8CdG0zPs1r1HE-VAklqP34DEmzHQ1DtuwMyIcduegaInlbXXNHF3wF0K5WvmeMNZKXQA42AwaiO2mvRbx9HF0aIp3g0OGO_bDidLmWKNAVaoZu4spL46Ao2zoRvfbTH0/s1600/IMG_1863.JPG" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="400" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjCC-Rt8XPrvBF8CdG0zPs1r1HE-VAklqP34DEmzHQ1DtuwMyIcduegaInlbXXNHF3wF0K5WvmeMNZKXQA42AwaiO2mvRbx9HF0aIp3g0OGO_bDidLmWKNAVaoZu4spL46Ao2zoRvfbTH0/s400/IMG_1863.JPG" width="397" /></a></div>
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I like how it looks from the top - like some 12 pointed star or something. </div>
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Amyhttp://www.blogger.com/profile/14741075245934673435noreply@blogger.com0