The Dice Lab, Sum of Cubes, and Double Polyhedra

Welcome to this week’s Math Munch!

It’s the final Thursday of September, so it’s time again for a recap of the month’s best from our Facebook page. This month we have a new sort of dice, a beautiful illustration of a numerical fact, and some wonderful new sculpture work from Rinus Roelofs. Let’s dig in.

First, check out this wonderful image. Meditate on it, and see if you can figure out what’s going on, even if you can’t understand the notation.

sum-of-cubes

It’s showing us a simple way to compute a sum of cubes. They can be broken down and reconstructed as a square! Consider the sum of the first 3 cube numbers, for example: 1+8+27=36, and 36 is the square of 6. One step further, 6 is the sum of the first 3 numbers.

So in the picture above, the sum of the first 5 cubes is equal to the square whose side length is the sum of 1 through 5.  AMAZING, and a beautiful illustration. Can you see why it always works, not just for 1 through 5? That’s key! And now test your understanding: What is the sum of the first 100 cube numbers?

dicelablogoUp next, we’ve met Henry Segerman plenty of times on Math Munch, including a look at the project he shares with Robert Fathauer, called The Dice Lab. They make mathematically interesting dice that have, in most cases, never been produced before. There newest creation (also last? see the video to see what I mean) is a 48-sided dice. Very cool. Can you think of a use for a 48-sided die?  It sure looks cool. Reminds me of a rhombic dodecahedron. Do you see the connection?

Finally, another familiar face – the incredible mathematical artist, Rinus Roelofs – has been making incredible things. We met Roelofs in July, but his facebook page has been full of activity since then. His recent work has focused on double-covered polyhedra.  You’ll have to click over and browse to see what I mean. Recently he posted a project I might want to take on. These are fold-up models for his creations. Check out the gallery below.

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I’m not 100% sure how that cube one works, but I think I can figure it out, and I bet some of you can too. Of course, I’m sure we’ll make mistakes, but if we keep on learning, I bet we can get this figured out. If anyone ends up making a template of their own, email it to us and we’ll share it on the site.

Until next time, bon appetit!

 

Demonstrations, a Number Tree, and Brainfilling Curves

This month of September has five Thursdays in it, so enjoy this bonus blast from the past. We hope it will “fill your brain”!

Math Munch

Welcome to this week’s Math Munch!

Maybe you’re headed back to school this week. (We are!) Or maybe you’ve been back for a few weeks now. Or maybe you’ve been out of school for years. No matter which one it is, we hope that this new school year will bring many new mathematical delights your way!

A website that’s worth returning to again and again is the Wolfram Demonstrations Project (WDP). Since it was founded in 2007, users of the software package Mathematica have been uploading “demonstrations” to this website—amazing illuminations of some of the gems of mathematics and the sciences.

Each demonstration is an interactive applet. Some are very simple, like one that will factor any number up to 10000 for you. Others are complex, like this one that “plots orbits of the Hopalong map.”

Some demonstrations are great for visualizing facts about math, like these:

Any Quadrilateral Can…

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Functionized Photos, Projective Games, and Traffic

Welcome to this week’s Math Munch!

Have you ever looked in a distorted mirror– one that stretched and squeezed your face so that you looked very, very silly? If you like that, check out this program called the Function Explorer that distorts your picture according to different functions!

Crazy Mikos

My cat under the “fraction” function

To use the program, you’ll have to turn on your webcam. Then, select one of the functions listed– maybe similarity, log, or fraction. Then, watch as the image in front of your webcam twists, expands, and repeats as the function distorts the picture!

What’s going on here? The program treats your picture like it’s on something called the complex plane— which is kind of like the regular two-dimensional plane we’re used to, except that some of the numbers multiply strangely. One of the dimensions on the complex plane is made of regular, normal numbers– which, in this situation, are called the “real numbers”– while the other dimension is made of different numbers, called “imaginary numbers.” These are the numbers that do weird things when you multiply them together. Maybe you’ve heard that you can’t take the square-root of a negative number. Well, on the complex plane you can. And when you do, you get an imaginary number!

Windows

Windows, under 1/z

If you’re curious about these crazy creatures called imaginary numbers and how they work to make images go wild on the complex plane, I recommend you check out this site. It gives a great interactive explanation of imaginary numbers (and teaches you about fractals, too!). But I also wouldn’t blame you if you wanted to spend a few hours holding things in front of your webcam and seeing what happens to them under different function transformations!

Gummy bears

Gummy bears! Which function did this?

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Meet Donna

Next up, I’d like to share a fun collection of games with you. They’re all made by mathematician Donna Dietz, and they all have to do with a particular kind of math that I find very interesting– projective geometry! You can still enjoy the games even if you know nothing about projective geometry (and you might learn something at the same time).

screen-shot-2016-09-14-at-9-19-29-pmThe rules are pretty simple: Donna gives you a bunch of cards with symbols on them. For example, in the version shown here, you get 13 cards with 4 symbols on them each. There are a bunch of different symbols. Your task is to pick four cards to discard and arrange the remaining nine so that the cards in each row, column, and diagonal share exactly one symbol.

Donna’s projective geometry games page has links to lots more games (if you think the game with cards in three rows and columns is too easy, try one with five) and information about them.

“What does this have to do with geometry?” you might be wondering. These games show a very important property of points and lines in projective geometry. In regular geometry (which you could also call Euclidean geometry), you can have two lines that don’t share any points– meaning that they’d be parallel. But this isn’t possible in projective geometry. All pairs of lines share exactly one point. How is this related to Donna’s games? If lines are rows, columns, and diagonals of cards, and points the symbols on them…

If you’d like to learn more about how and why Donna developed these games, check out this page!

Finally, I’ve been driving a lot lately. I live in the Bay Area, and there is SO MUCH TRAFFIC AAAAAAAA!!! I went searching for solutions, and I came across this great video by our friend CGP Grey (who also made these great videos about voting theory). There’s a lot of math going on here, even if it isn’t immediately apparent. Can you find the math? (Oh, and can you stop causing traffic jams? Thanks.)

Don’t Math Munch and drive, and bon appetit!

Sphericon, National Curve Bank, and Cardioid Art

Check out this re-run post from March, 2016! I still can’t get enough of Sphericons… Enjoy!

Welcome to this week’s Math Munch!

Behold the Sphericon!

What is that? Well, it rolls like a sphere, but is made of two cones attached with a twist– hence, the spheri-con! The one in the video is made out of pie (not sure why…), but you can make sphericons out of all kinds of materials.

Wooden sphericonIt was developed by a few people at different times– like many brilliant new objects. But it entered the world of math when mathematician Ian Stewart wrote about it in his column in Scientific American. The wooden sphericon was made by Steve Mathias, an engineer from Sacramento, California, who read Ian’s article and thought sphericons would be fun to make. To learn more about how Steve made those beautiful wooden sphericons, check out his site!

Even if you’re not a woodworker, like Steve, you can still make your own sphericon. You can start with two cones and make one this way, by attaching the cones at their bases, slicing the whole thing in half, rotating one of the halves 90 degrees, and attaching again:How to make a sphericon

Or you can print out this image, cut it out, fold it up, and glue (click on the image for a larger printable size):

Sphericon pattern

If you do make your own sphericon (which I recommend, because they’re really cool), watch the path it makes as it rolls. See how it wiggles? What shape do you think the path is?

ncbmastertitleI found out about the sphericon while browsing through an awesome website– the National Curve Bank. It’s just what it sounds like– an online bank full of curves! You can even make a deposit– though, unlike a real bank, you can take out as many curves as you like. The goal of the National Curve Bank is to provide great pictures and animations of curves that you’d never find in a normal math book. Think of how hard it would be to understand how a sphericon works if you couldn’t watch a video of it rolling?

epicycloidaThere are lots of great animations of curves and other shapes in the National Curve Bank– like the sphericon! Another of my favorites is the “cycloid family.” A cycloid is the curve traced by a point on a circle as the circle rolls– like if you attached a pen to the wheel of your bike and rode it next to a wall, so that the pen drew on the wall. It’s a pretty cool curve– but there are lots of other related curves that are even cooler. The epicycloid (image on the right) is the curve made by the pen on your bike wheel if you rode the bike around a circle. Nice!

You should explore the National Curve Bank yourself, and find your own favorite curve! Let us know in the comments if you find one you like.

String cardioid

String art cardioid

Finally, to round out this week’s post on circle-y curves (pun intended), check out another of my favorite curves– the cardioid. A cardioid looks like a heart (hence the name). There are lots of ways to make a cardioid (some of which we posted about for Valentine’s Day a few years ago). But my favorite way is to make it out of string!

String art is really fun. If you’ve never done any string art, check out the images made by Julia Dweck’s class that we posted last year. Or, try making your own string art cardioid! This site shows you how to draw circles, ovals, cardioids, and spirals using just straight lines– you could follow the same instructions, replacing the straight lines you’d draw with pieces of string attached to tacks! If you’re not sure how the string part would work, check out this site for basic string art instructions.

Bon appetit!

Squricangle, Magic Angle Sculpture, and …

Welcome to this week’s Math Munch!

There’s a neat old problem/puzzle that goes like this: make a 3-D shape that could fit snugly through each of three holes—one a square, one a circle, and one a triangle. To make a shape that works for just two holes isn’t so tricky. For example, a cylinder that is just as tall as it is across would fit snugly through a circle hole and a square hole. Can you think of what would work for each of the other two shape combos? What about all three?

wedge-holes

Three holes, three shapes…and what’s that over in the corner??

If you’re curious about the answer, you might enjoy this post by Kit Wallace or this page by George Hart or—believe it or not—roundsquaretriangle.com. I don’t know the origin of this puzzle and would love to. I haven’t found any info about it after to poking around the internet for a while. So if you locate any information about the backstory of the squircangle—which is not its real name, just one that I made up—please let us know!

Even though I knew about the square-circle-triangle problem, I was not at all prepared to encounter the solution to the jet-butterfly-dragon problem!

dbj

Dragon Butterfly Jet is just one of several “magic angle sculptures” created by artist, chemist, and PhD, and high school dropout John V. Muntean. John writes the following in his Artist Statement:

As a scientist and artist, I am interested in the how perception influences our theory of the universe. … Every 120º of rotation, the amorphous shadows evolve into independent forms. Our scientific interpretation of nature often depends upon our point of view. Perspective matters.

There’s much more to see on John’s website. And you can check out Dragon Butterfly Jet in action in the video below, along with Knight Mermaid Pirate-Ship. I also recommend this video made by John where he demonstrates how his sculpture works himself. It also includes a stop-frame animation of the sculpture being built! So cool.

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No, not ellipses…

And finally, what you’ve all been waiting for…

…!

That’s right! My final share of the week is that most outspoken of punctuation marks, the ellipsis. Because often what you don’t say says a whole lot! That’s true when writing a story or some dialogue, and it’s also true in mathematics. Watch: 1+2+3+…+100. See? Pretty neat! Those three dots sure say a mouthful…

The ellipsis is probably my second favorite punctuation mark—after the em dash, of course. But don’t take my word for it. Instead, check out this article about the history and uses—mathematical and otherwise—of the humble ellipsis. Author Cameron Hunt McNabb writes:

Thus the ellipsis has been used to indicate anything from the erroneous to the irrational, and its intrigue lies in resistance to meaning. As long as we have things to say, we will have things to omit.

witte

The very first equals sign, in 1557.

I could go on and on about the ellipsis, just like pi does: 3.1415… But anyway, while we’re on the subject of punctuation, let me point you to one of my favorite sites on the mathematical internet: the Earliest Uses of Various Mathematical Symbols page, maintained by Jeff Miller. Jeff teaches high school math in Florida and also has some other great pages, too, including this one about mathematicians featured on stamps.

Bon…

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A nice visualization of the squircangle by Matt Henderson

…appetit!

Squaring, Water Calculator, and Snap the Turtle

Welcome to this week’s Math Munch!

I’ve been really into squares lately. Maybe it’s because I recently ran across a new puzzle involving squares– something called Mrs. Perkin’s quilt.

Mrs. Perkin's quilt 1

69 by 69 Mrs. Perkin’s quilt.

The original version of the puzzle was published way back in 1907, and it went like this: “For Christmas, Mrs. Potipher Perkins received a very pretty patchwork quilt constructed of 169 square pieces of silk material. The puzzle is to find the smallest number of square portions of which the quilt could be composed and show how they might be joined together. Or, to put it the reverse way, divide the quilt into as few square portions as possible by merely cutting the stitches.”

Mrs. Perkin's quilt 18

18 by 18 Mrs. Perkin’s quilt

Said in another way, if you have a 13 by 13 square, how can you divide it up into the smallest number of smaller squares? Don’t worry, you get to solve it yourself– I’m not including a picture of the solution to that version of the puzzle because there are so many beautiful pictures of solutions to the puzzle when you start with larger and smaller squares. Some are definitely more interesting than others. If you want to start simple, try the 4 by 4 version. I particularly like the look of the solution to the 18 by 18 version.

Mrs. Perkin's quilt 152

152 by 152 Mrs. Perkin’s quilt

Maybe you’re wondering where I got all these great pictures of Mrs. Perkin’s quits. And– wait a second– is that the solution to the 152 by 152 version? It sure is– and I got it from one of my favorite math websites, the Wolfram Demonstrations Project. The site is full of awesome visualizations of all kinds of things, from math problems to scans of the human brain. The Mrs. Perkin’s quilts demonstration solves the puzzle for up to a 1,098 by 1,098 square!

Next up, we here at Math Munch are big fans of unusual calculators. Marble calculators, domino calculators… what will we turn up next? Well, here for your strange calculator enjoyment is a water calculator! Check out this video to see how it works:

I might not want to rely on this calculator to do my homework, but it certainly is interesting!

Snap the TurtleFinally, meet Snap the Turtle! This cute little guy is here to teach you how to make beautiful math art stars using computer programming.

On the website Tynker, Snap can show you how to design a program to make intricate line drawings– and learn something about computer programming at the same time. Tynker’s goal is to teach kids to be programming “literate.” Combine computer programming with a little math and art (and a turtle)– what could be better?

I hope something grabbed your interest this week! Bon appetit!

Lucea, Fiber Bundles, and Hamilton

Welcome to this week’s Math Munch!

The Summer Olympics are underway in Brazil. I have loved the Olympics since I was a kid. The opening ceremony is one of my favorite parts—the celebration of the host country’s history and culture, the athletes proudly marching in and representing their homeland. And the big moment when the Olympic cauldron is lit! This year I was just so delighted by the sculpture that acted as the cauldron’s backdrop.

Isn’t that amazing! The title of this enormous metal sculpture is Lucea, and it was created by American sculptor Anthony Howe. You can read about Anthony and how he came to make Lucea for the Olympics in this article. Here’s one quote from Anthony:

“I hope what people take away from the cauldron, the Opening Ceremonies, and the Rio Games themselves is that there are no limits to what a human being can accomplish.”

Here’s another view of Lucea from Anthony’s website:

Lucea is certainly hypnotizing in its own right, but I think it jumped out at me in part because I’ve been thinking a lot about fiber bundles recently. A fiber bundle is a “twist” on a simpler kind of object called a product space. You are familiar with some examples of products spaces. A square is a line “times” a line. A cylinder is a line “times” a circle. And a torus is a circle “times” a circle.

squarecylindertorus

Square, cylinder, and torus.

So, what does it mean to introduce a “twist” to a product space? Well, it means that while every little patch of your object will look like a product, the whole thing gets glued up in some fancy way. So, instead of a cylinder that goes around all normal, we can let the line factor do a flip as it goes around the circle and voila—a Mobius strip!

cylinder-mobius

Now, check out this image:

lBUHy

It’s two Mobius strips stuck together! Does this remind you of Lucea?! Instead of a line “times” a circle that’s been twisted, we have an X shape “times” a circle.

Do you think you could fill up all of space with an infinity of circles? You might try your hand at it. One answer to this puzzle is a wonderful example of a fiber bundle called the Hopf fibration. Just as you can think about a circle as a line plus one extra point to close it up, and a sphere as a plane with one extra point to close it up, the three-sphere is usual three-dimenional space plus one extra point. The Hopf fibration shows that the three-sphere is a twisted product of a sphere “times” a circle. For a really lovely visualization of this fact, check out this video:

That is some tough but also gorgeous mathematics. Since you’ve made it this far in the post, I definitely think you deserve to indulge and maybe rock out a little. And what’s the hottest ticket on Broadway this summer? I hope you’ll enjoy this superb music video about Hamilton!

William Rowan Hamilton, that is. The inventor of quaternions, explorer of Hamiltonian circuits, and reformulator of physics. Brilliant.

citymapHere are a couple of pages of Hamiltonian circuit puzzles. The goal is to visit every dot exactly once as you draw one continuous path. Try them out! Rio, where the Olympics is happening, pops up as a dot in the first one. You might even try your hand at making some Hamiltonian puzzles of your own.

Happy puzzling, and bon appetit!