From the Doodle Book III

"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." - Danica McKellar 

Another Proof Without Words

Here's another pretty famous proof without words - it shows that the sum of the first n odd integers is n². 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 2² which is 4 beads in total. Adding the 3rd odd integer 5, we get 3² = 9 beads. 

Said another way, 

1=1=1²

1+3=4=2²
1+3+5=9=3²
1+3+5+7=16=4²
1+3+5+7+9=25=5²

You could realize this via algebra as well. Consider the two consecutive square numbers n² and (n+1)². Expanding (n+1)², you get  n²+2n+1. Then the difference between one square number n² and the next square number (n+1)² is simply (n²+2n+1)-n² = 2n+1. So to get from the nth square number to the next square number, you just add the next odd number, 2n+1.

Apples and Solids of Revolution

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. 

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. 

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. 

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). 

Conceptual drawing of an apple as approximated by cylindrical disks

In the end, the approximate volumes I got were as follows: 

1 cut (2 disks) - 171.05972 cm³ 

2 cuts (3 disks) - 165.95463 cm³

3 cuts (4 disks) - 157.76031 cm³

4 cuts (5 disks) - 126.77023 cm³

The approximations nicely approach the true volume of 121 cm³. 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. 

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. 

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. 

A Little Bit of Binary

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. 

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) 

Step 1: 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. 

Step 2: 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. 

Step 3: 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).

Cuboctahedron from Magnets

I made something today. After some internet searchings, I discovered that this is called a cuboctahedron. 

It is quite pretty and symmetric. The red pieces show that it can be made of pyramids and tetrahedrons.

Proof Without Words

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? 

A Study in Orange

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...

Upcycled Bottle Top Ring Construction

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.

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.

A shot from the top.

Optimization Problem via Geometry

So I was browsing my old calculus book (don't judge...yes, I browse my calculus book), and I saw this optimization problem -

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 θ1=θ2.

I'm pretty sure I even did this question, but now looking back on it the answer seems obvious. 

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 θ1 and θ2 are vertical angles so they're congruent. 

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. 

From the Doodle Book II

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.

Kabob Skewer Hyperboloid

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. 

 An alternate view

I like how it looks from the top - like some 12 pointed star or something. 

Pretty Paper Polyhedra

The five Platonic solids made from index cards. I suppose the Platonic solids are just innately beautiful in their symmetry - look how perfect and congruent they are!

Source

Did you know that the five Platonic solids were associated with the elements? There is actually rationale behind these associations - the tetrahedron is fire because of the sharpness of its edges and vertices, the stable cube is earth, water flows out of one's hand like an icosahedron might, between the tetrahedron (fire) and the icosahedron (water) is the intermediary octahedron (air). I particularly like the dodecahedron's representation of the universe - Plato said, "There remained a fifth construction which God used for embroidering the constellations on the whole heaven." 

Source

Another particularly awesome bit of Platonic solid history is when astronomer Johannes Kepler proposed that the spacing of the planets could be explained with a series of nested polyhedra and spheres. Starting with the octahedron, followed by the icosahedron, dodecahedron, tetrahedron, and finally the cube, this elegant geometric arrangement was said to explain the locations of the six known planets (Mercury, Venus, Earth, Mars, Jupiter, Saturn). Though the theory had to be abandoned eventually, the research that Kepler did led to his discovery that the planets have elliptical rather than circular orbits. 

A Hexaflexagon Obsession

My hexaflexagon collection. If you don't know what a hexaflexagon is, I demand you go and watch all of these - link.

Fun things to do with flexagons -

• Cut them in half as you would a Mobius strip. No seriously. In the picture to the left of the giant yellow one, you can kind of see what happens - with it's one side one edge awesomeness, it's kind of like this three twist Mobius strip that's been squashed flat all symmetrically! 
• Decorate them with cool patterns and colors - I've seen some very cool Escher flexagons (but I mean I have a huge Escher soft spot...) 
• 3D flexagons? I think yes. 

From the Doodle Book I

Sharpies + highlighters 

Fractal-esque Color Wheel

Color wheel I designed and painted based on a fractal called an Appollonian gasket, seen here. The fractal is named after the Greek mathematician Appollonius of Perga (also the guy who gave the ellipse, parabola, and hyperbola their names!) 

It was quite a lot of fun to construct - I did the equilateral triangle, the largest incircle, and the three largest primary and secondary color circles with traditional construction (straightedge and compass - that was quite fun to figure out). Everything else was done freehand mainly because they were getting so small that constructing them became really difficult. 

The Appollonian gasket is generated from triplets of circles, where each circle is tangent to the other two, so the three starting circles were the primary colors (red, yellow, blue) which then branched off to secondary colors (green, orange, purple) and finally tertiary colors (yellow+orange=amber, orange+red=vermillion, red+purple=magenta, purple+blue=violet, blue+green=viridian, green+yellow=chartreuse). I really like the idea of a fractal-esque color wheel - I feel like it gives the usual circular repeating concept of color wheels a new meaning because it repeats forever, continuing on without end. 

If I were any good at programming I would love to do one on the computer and make all the circles and color mixings super precise, going on to do the quaternary and quinary colors too...what's particularly symbolic is that these colors go on to grays and browns that approach but never reach black, echoing back to their fractal-esque, infinite nature.

Optimal Bubbleology

This beautiful shape is simply the result of dipping a tetrahedral wire frame into soapy water. As you've probably noticed before, ordinary soap bubbles are round, but have you ever wondered why? Soap bubbles will assume the shape of least surface area possible to contain a given volume - this is known as a minimal surface. In a bubble, the inward surface tension forces of the water film are exactly balanced by the outward pushing pressure of the air inside.

Because of this lovely property, we can find these gorgeous and ephemeral solutions to problems in calculus of variations of finding the minimal surface of a boundary with specified constraints. This is known as Plateau's Problem, raised by Joseph-Louis Lagrange in 1760 but named after the Belgian physicist Joseph Plateau who solved some special cases of the problem experimentally using soap films and wire frames.

This one is a cube frame. Plateau noticed that only three smooth surfaces of a soap film can meet along a line (called a Plateau border) and the angle between any two of these surfaces is 2π/3 radians. In addition, only four Plateau borders may meet at a point and must meet at tetrahedral angles (arccos(1/3)) .

If you take the resulting minimal surface and pop several of the boundaries, you can get some really neat surfaces - here's one from the cube that turns out to have a saddle point.

Another great one is adding a bubble to the center of the cube, which gives a three dimensional representation of a 4D hypercube!

“Nature is written in that great book which ever is before our eyes -- I mean the universe -- but we cannot understand it if we do not first learn the language and grasp the symbols in which it is written. The book is written in mathematical language, and the symbols are triangles, circles and other geometrical figures, without whose help it is impossible to comprehend a single word of it; without which one wanders in vain through a dark labyrinth.” - Galileo Galilei