Author Archives: Paul Salomon

Near Miss, Curiosa Mathematica, and Poincaré

Welcome to this week’s Math Munch!

For this last Thursday of April, we’ll be taking a look at some recent posts from our facebook page. Craig Kaplan writes about “near miss” polyhedra, a Pythagorean gif takes us to an curious math blog, and we find a beautiful portrait of a great mathematician.  Let’s go!

Craig Kaplan

Craig Kaplan

First is an article from a wonderful mathematician and mathematical artist by the name of Craig Kaplan. His name has popped up on Math Munch before (1, 2 ,3), in case it sounds familiar. You can check out Craig’s stuff on his website, Isohedral, or download his really great game, “Good Fences,” which I have on my iPhone.

near missWhat I really wanted to share, however was Craig’s writing on “A New Near Miss.” This is a polyhedron that almost is… but just isn’t. It looks pretty good, but it can’t be. You’ll have to read to see what I mean.

PythagorasPerigalP.gifUp next, I found this little gif on our facebook page, and I absolutely loved it. It demonstrates the Pythagorean Theorem which says that as long as that’s a right triangle there, the big square on bottom is exactly as big as the two smaller squares combined. The animation shows you how to chop up the middle-sized square and recombine it with the small one to make the big one. I knew there were demonstrations/proofs like this, but this one opened my eyes to something I didn’t quite know before.

This gif sent me off on a journey through the internet to track down the source, and it led me to a site called Curiosa Mathematica. It’s a math blog featuring lots of random math goodies. There’s lots to see and get into (much like Math Munch). Here’s a quote I found there.  I hope you find something you like too.

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Finally, I was really taken by this piece of art (below). It’s a portrait of French mathematician Henri Poincaré, and it was drawn by Bill Sanderson. I can’t find much info on Bill, but WOW the piece is so cool. I love how he’s surrounded by his mathematical creations. I was hoping he had done more, and I did find a couple more (below), but not all I had hoped for.

 

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French Mathematician Henri Poincaré

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Alan Turing

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Isaac Newton

Have some illustrative talent? I’d love to see your mathematician’s portrait. Feel free to send us something… anything.

I hope you enjoy your weekend and find something tasty out there in the mathematical interwebs. Bon appetit!

2016, ScienTile, and a New Algorithm

Welcome to this week’s Math Munch!

In this week’s post we check out a tile designing contest from 2010, learn about some breakthrough news in computational algorithms, and get a DIY project to ring in the new year.

Speaking of the new year… welcome to the new year!! 2016 is 11111100000 in binary, by the way. Pretty cool right!? The five 0’s at the end tell you that 2016 has five 2’s in its prime factorization. That is, you can divide 2016 by 2 five times and still get a whole number. The big bunch of 1’s at the start means its also divisible by a number that is one less than a power of 2. 63, basically.

That is to say, 2016 = (26–1)(25). I think that signals a promising year. Bring it on.

2016 card project

DIY Möbius strip project to ring in the new year

cord project template

template here

Here’s a great way to start your year off. How about a paper folding project from mathematical artist Rinus Roelofs? I found the project posted by imarginary.org on their facebook page. According to the post, “this card is the representation of a Möbius strip.”

Click here for the downloadable, printable, cuttable, foldable, template and make your own!

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A tile pattern I designed. Anyone else thinking bee hive?

Up next, I’ve been playing a lot of board games that use square or hexagon tiles, and I’ve been thinking about what other tiles might make for a cool game. First off, here’s a little tile I came up with that always leaves hexagons in the places where they meet. Might make a neat game  where you build a bee colony. Who knows.  But in my searching for groovy tiles, I found ScienTile.

ScienTile was an “open tile design competition” initiated by Dániel Erdély, a Hungarian mathematician and mathematical artist featured previously on MM for his spidrons.  In fact, ScienTile was meant to commemorate the 2010 Bridges Conference, which was in Hungary. Sadly, I don’t think the ScienTile competition was repeated in later years, but the results from 2010 are quite beautiful. I was most struck by the picture below, a tile designed by Gabor Gondos. I also really liked this one by the wonderful Craig Kaplan (featured previously here), but all the submissions can be found here.

A gorgeous and flexible tile design by Gabor Gandos

A gorgeous and flexible tile design by Gabor Gondos

All these graphs are isomorphic, and the new algorithm could tell you that... really fast!

All these graphs are isomorphic, and the new algorithm could tell you that… really fast!

Finally, some breakthrough math news from the computational world. Computer scientists have develop a new fast algorithm for solving” the graph isomorphism” problem, which simply checks whether or not two graphs (think connect-the-dots pictures) are really the same. All the graphs in the gif on the right are isomorphic, because they can be morphed into each other without changing the connectivity of the dots.

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The 5,2 Johnson Graph

The new algorithm breaks a computational record that was unbroken for the last 30 years, which is a crazy long time in computer terms. Congratulations László Babai, who can be seen below presenting his breakthrough paper at the University of Chicago. His algorithm actually doesn’t cover all types of graphs, but Babai was able to show that the only type of graph not covered were the highly symmetric Johnson Graphs. You can see one of these on the right.

You can find more info on the record-breaking algorithm in this article from ScienceNews.org or this write up from Quanta Magazine.

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László Babai presenting his record-breaking algorithm

Have a great week, and bon appetit!

The Colorspace Atlas, allRGB, and Hyperbolic Puzzles

Welcome to this week’s Math Munch!

Update: A few weeks ago we met Dearing Wang, mathematical artist and creator of Dearing Draws. Now you can read a Math Munch Q&A with Dearing Wang.

OK, first up in this week’s post, do you remember when we talked about the six dimensions of color and the RGB color system? Well either way, consider this:

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Artist Tauba Auerbach (one of my absolute favorite contemporary artists) made a book that contains every possible color!!! Tauba calls it “The RGB Colorspace Atlas.” The book is a perfect 8″ by 8″ by 8″ cube, matching the classic RGB color cube.

RGB_Cube_Show_lowgamma_cutout_aThe primary colors of light (red, blue, and green) increase as you move in each of the three directions. This leaves white and black at opposite corners of the cube, and all the wonderful colors spread around throughout the cube, with the primary and secondary colors on the other corners. You can read more here, if you like.

The book shows cross-sections moving through a single axis, so Tauba really had 3 choices for how the pages should flip through the cube. In fact, she made all three books!  Jonathan Turner made simulations of all three axes however, so we can see each one if we like. Can you tell which one is open in the pictures above?

That’s the Red Axis. Compare that to the Green Axis and Blue Axis.

For computer graphics, RGB color codes are ordered triples of numbers like (120, 15, 28). Each number says how much of each color should be included in the mix.  There are 256 possible values for each one, with values from 0 to 255. [Examples: (0,0,0) is black. (255,255,255) is white.  (255,0,0) is red. (127,0,0) is a red that’s half as bright.] Since there are only so many number combinations, computers have exactly 16,777,216 possible colors. That’s where allRGB comes in.

starry-night

Starry Night

hilbert

Hilbert Coloring

escher-reptiles

Escher LIzards

As they say, “The objective of allRGB is simple: To create images with one pixel for every RGB color (16777216); not one color missing, and not one color twice.” AllRGB is a bounded concept, since there are only finitely many ways to rearrange those 16777216 pixels. But of course there are a HUUUUGGGEEEE number of ways to rearrange them, so there’s lots to see. (In fact if you wrote a 1 with 100 million zeroes after it, that number would still be smaller than the number of allRGB pictures!! And that’s only part of the story)  Click the pictures above for zoomable versions as well as descriptions of their creation.

hyperbolic maze 1 hyperbolic maze

We’ve posted a little before about hyperbolic geometry. Very very briefly, the hyperbolic plane is a 2D surface where some of our usual intuition gets a little warped. For example, two lines can be parallel to the same line but not parallel to each other, which seems a little awkward. Click the images above to really experience what it’s like to walk through a hyperbolic world. David Madore created these hyperbolic “mazes,” which give you a birds eye view as you walk through a strange new land.

Finally, you might enjoy this old Numberplay puzzle with a hyperbolic feel, based on the movements of whales.

Gary Antonick asks "What is the fewest-bun path between the two white buns? (The two white buns are the first and last — or 40th — buns in the top row."

Gary Antonick asks “What is the fewest-bun path between the two white buns? (The two white buns are the first and last — or 40th — buns in the top row.”

What do buns have to do with whales and hyperbolic geometry? You’ll just have to click and find out.

Have a great week and bon appetit!