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If the Earth were a doughnut

The famous philosopher Charles Saunders Peirce worked for a while at the U. S. Coast and Geodetic Survey. He invented a new map of the Earth. First you map the spherical Earth to a torus. Then you slice open the torus and unroll it to a square.
You get this:

It's called Peirce's quincuncial.

Puzzle: why is it called that?

If you then make wallpaper with such squares, you get a cool infinite repeating map of the Earth, shown here:

These pictures are by Carlos A. Furuti.

But +Greg Egan did something cooler. He made an animation of the torus, showing how the continents move on it as the spherical Earth turns!

The math here is the math of elliptic functions. I explained it here:

As the world turns, but the map stays still

The animation below was suggested by Michael Hardy, whose question on MathOverflow about the properties of a 2-to-1, almost-everywhere-conformal map from the torus to the sphere got me interested in the subject.

In a previous post, I showed a fixed mapping of two copies of the surface of the Earth onto a torus, and then rotated the resulting torus to reveal more of its surface than can be seen in any one fixed view.

But in this animation, while the torus itself stays fixed, the Earth rotates around its axis relative to the coordinate system used for the mapping.

The result makes it clear just how strange things are at some points on this map (in the previous view, the strangeness was deliberately hidden in the ocean). There are four branch points on the torus; if you walk around these points, the version of you mapped to the Earth will complete two circles around the corresponding point. You can only see two of the branch points in this view, but the positions of the other two are easy to imagine from the symmetry.

[Edited to add] Another version of this image (with a grid of longitude and latitude marked, and twice as many frames to give a smoother rotation – the file is about 17 Mb):

My previous post on the torus map:

Michael Hardy's question on MathOverflow:

+Henry Segerman's interactive 3D model (which uses essentially the same map, but orients the Earth differently relative to the torus):

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How to fairly share a square cake among 5 people

Suppose you have a square cake of arbitrary height and want to divide it into 5 pieces that all have the same amount of cake and the same amount of icing.

The icing makes it hard.  If there were no icing on the cake, or only icing on top, we could cut the cake in 5 strips of equal thickness. 

But let's assume there's icing on the sides of the cake too!  Since we don't know how tall the cake is, we want to slice the cake vertically into pieces that have equal area on top and contain equal amounts of the outside edge. 

This solution by Tim, a math teacher in Wisconsin, is quite impressive.  Divide each side into 5 parts as shown and cut straight to the center of the cake at C. 

Puzzle 1.  But is this solution correct?

You can see other answers here:

The question was raised by Steven Strogatz on Twitter, and I heard about it from +Alok Tiwari, who heard about it from +Ian Agol.  Some of the answers on the original Twitter thread are really dumb, some are really smart.  It's fun to see them all.

The fun, of course - let's come out and say it! - arises from the gnarly and complicated relationship that the numbers 4 and 5 have with each other.  Squares and regular pentagons don't play nicely, and here Tim is trying to pentisect the square.

Puzzle 2. What's the easiest way to construct a segment of length sqrt(5/4) using ruler and compass?


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The Chmutov octic surface

You can get some very fancy surfaces using just polynomial equations. Here +Abdelaziz Nait Merzouk drew one using polynomials of degree 8.  That's why it's called an octic.  

Why is it called the Chmutov octic?   Well, that's because it was constructed by V. S. Chmutov as part of an effort to build surfaces with lots of ordinary double points, meaning points that look the place where the tips of two cones meet.  This one has 144 ordinary double points!

That's not the best you can do: the octic with the highest known number of ordinary double points is the Endrass octic, shown here:

The Endrass octic has 168 ordinary double points.  Nobody knows if that's the best possible.

The Chmutov octic is just one of a series of surfaces invented by Chmutov.  There's a Chmutov quadratic, a Chmutov cubic, a Chmutov quartic, a Chmutov quintic,  a Chmutov sextic, a Chmutov septic, a Chmutov octic, a Chmutov nonic, a Chmutov decic, a Chmutov hendecic, a Chmutov duodecic, a Chmutov triskaidecic, a Chmutov tetrakaidecic, a Chmutov pendecic, a Chmutov hexadecic, a Chmutov heptadecic, a Chmutov octadecic, a Chmutov enneadecic, a Chmutov icosic, and so on.  In fact you can see a quick animated gif of all of these - from the quadratic to the icosic - here:

Again, it was made by +Abdelaziz Nait Merzouk.  You'll notice that most of the Chmutov surfaces of even degree look a lot like the octic here, while those of odd degree extend out to infinity. 

Chmutov made these surfaces to get a lower bound on how many ordinary double points we could cram into a surface of a given degree.  In most cases other people have beaten him by now.  But still, these surfaces are cute!  They're defined using some polynomials invented by the Russian mathematician Chebyshev - also known as Chebychev, Chebysheff, Chebychov, Chebyshov, Tchebychev, Tchebycheff, Tschebyschev, Tschebyschef, or Tschebyscheff.  Apparently he suffered from a rare psychological disorder that made him forget how to spell his name - so each time he wrote another paper, he signed it a different way!

Happy New Year!  (You may not have heard, but this year April Fool's Day has been scheduled on January 1st instead of April 1st.)


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Romik's ambidextrous sofa

The ambidextrous moving sofa problem is to find the planar shape with the biggest area that can slide through right-angled turns both to the right and to the left in a hallway of width 1.  

Earlier this year Dan Romik, a mathematician at the University of California Davis, found the best known solution to this problem!   He created this animated gif of it, too.  His shape is bounded by 18 curves, each of which is either part of a circle, or part of a curve described by a polynomial equation of degree 6.   

Nobody has proved his solution is optimal.   We're not even sure that it's locally optimal, meaning that you can't make slight changes in his shape that increase the area and get a shape that still fits down the hallway.  This is an interesting challenge.

For more, including the precise area of this shape, try my blog article on Visual Insight:

I hope you're all having a great holiday!

Each year I try to think of things I can stop doing... so I can do more new stuff.   In 2017, I will try to take a year-long break from posting articles on Visual Insight.  I've been doing two a month for quite a while, I've done 81 of them, and I'm running out of enthusiasm.  Also, right now, a lot of my energy is going into the Azimuth Backup Project.  So, maybe I will save up ideas and restart Visual Insight in 2018.  But perhaps I'll end with a bang on January 1st, 2017.

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Truncated hexagonal tiling honeycomb

+Roice Nelson has been drawing more honeycombs!  This is one of my favorites.  To get it, you start with the hexagonal tiling honeycomb.  That's is a way of putting lots of sheets tiled by hexagons into hyperbolic space, a curved 3-dimensional space.   It's easier to understand with a picture, so look at the second one here:

Starting from this, you take each place where 4 edges meet and replace it with a little tetrahedron.  That gives you the truncated hexagonal tiling honeycomb, shown here.  Beautiful!

There's a limited collection of structures this nice, and mathematicians have classified them.  A classification theorem lets you survey the options: it's like a mineral collection.


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Alien machinery

That's what it looks like to me.  But it's an image created by Greg Egan, the science fiction author.   And there's a story behind it.

Egan and I figured out a bunch of stuff about the McGee graph, a highly symmetrical graph with 24 vertices and 36 edges.   I wrote an article about it on Visual Insight, my blog for beautiful math pictures. 

Later I got an email from Ed Pegg, Jr saying he'd worked out a unit-distance embedding of the McGee graph: a way of drawing it in the plane so that any two vertices connected by an edge are distance 1 apart.  He wanted to know if this was rigid or flexible.  In other words, he wanted to know whether you can change its shape slightly while it remains a unit-distance embedding.

Egan thought about it a lot and did a lot of computations and discovered that this unit-distance embedding is flexible.  And here it is, flexing!

For Pegg and Egan's work, go here:

What's the practical use of all this?  Mainly, it's a practice problem in structural rigidity: the study of whether a structure is flexible or rigid.  This is important in engineering:

A structure is infinitesimally flexible if, roughly, we can bend it a teeny weeny bit.  As the name suggests, infinitesimal rigidity can be determined by using calculus to take the derivative of all the edge lengths as a function of all the vertex positions and then using linear algebra to see in which directions this derivative is zero.  This is easy in principle, though complicated when you have 24 vertices and 36 edges.

Puzzle 1: with a minimum of explicit computation, prove that any unit-distance embedding of the McGee graph is infinitesimally flexible.

Infinitesimal flexibility is a necessary but not sufficient condition for true flexibility.

Puzzle 2: find a unit-distance embedding of a graph that is infinitesimally flexible but not flexible.

So, Egan had to do more work to show Pegg's unit-distance embedding of the McGee graph was actually flexible.  There is probably a high-powered theoretical way to do this, and it's probably not even very complicated, but I don't know it.   Do you?

For my Visual Insight post on the McGee graph, go here:

By the way, I don't like the phrase 'unit-distance embedding' - we're not really embedding the McGee graph in the plane, because we're letting the edges cross.   The word 'immersion' would be better. 

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McGee graph

On MathOverflow, someone named მამუკა ჯიბლაძე made this cool animation of the McGee graph, which has 24 dots and 36 edges. 

This movie illustrates a symmetry of the McGee graph.  In other words, if you let the picture make a quarter turn, it looks just the same, even though the dots have moved. 

In fact, even if you let the graph make a full turn, the dots have moved from their original position!   Why?  Because the red edges have flipped upside down.  So, you need to let the graph make 2 full turns before everything returns to its original position.

So, this movie illustrates 2 × 4 = 8 symmetries of the McGee graph.  But the McGee graph actually has a total of 32 symmetries.  These symmetries are precisely the transformations of the "affine line over Z/8".  For details, try this:

Who is მამუკა ჯიბლაძე?  It's Mamuka Jibladze, writing in Georgian script.   There are lots of good category theorists from Georgia - not the southern state in the US, the country between Russia and Turkey.  Mamuka Jibladze is one.

Here is Jibladze's website:

Here is Jibladze's post on MathOverflow:

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Diamonds are forever?

This shows the pattern of carbon atoms in a diamond.  Each atom is connected to 4 neighbors.  Its neighbors are the corners of a regular tetrahedron! 

The mathematics of this pattern is beautiful, and I explain it here:

I also explain hyperdiamonds in 4 or more dimensions.  The hyperdiamond in 8 dimensions is especially awesome: it's called the E8 lattice, and it's connected to string theory, the octonions and more.  

In 1888, Cecil Rhodes started a company called De Beers to sell the diamonds dug up by slaves in Botswana, Namibia, and South Africa.  De Beers got a total monopoly on diamonds.  To keep the price up,  they wanted a slogan to make diamonds into the jewel of choice for weddings:

After unsuccessfuly trying to create a slogan for De Beers which would perfectly express of the qualities of a diamond mingled with romance, there was finally a stroke of genius in 1947. One of Ayer’s young copyrighters, Frances Gerety, was working late one night to the point of exhaustion. Finally in desperation, she put her head down on the table and pleaded for help. Just before she left work that night she scribbled the words “a diamond is forever” on a piece of paper and the rest is history. This may have had a simple start, but the result was America’s most famous advertising slogan and today over 90% of American’s recognize it.

Diamonds are very hard, but they don't actually last forever.  Very few things do, because forever is a very long time.  As far as we can tell, electrons are forever.   Many theories of particle physics predict that protons decay.  However, experiments indicate that they last for at least 10^34 years on average... so these theories have not been confirmed.  Protons may be forever.

Diamonds, on the other hand, are metastable at room temperature and pressure.  They are formed under high pressure, deep underground, but under ordinary conditions at the Earth's surface a diamond has more energy than the same amount of graphite!  So, it should slowly turn into graphite and release energy. 

So, diamonds are not forever.  However,  they last a very long time.

How about the abstract mathematical structure of the diamond.   Does that last forever?

Not really.   This is not a thing in spacetime, it's an abstract pattern that has the ability of being realized at any place, at any time, in any universe.  It doesn't make sense to say that this mathematical pattern lasts forever - or that it doesn't last forever.   "Lasting forever" applies to things in spacetime.

Puzzle: what is the approximate lifetime of a diamond at room temperature and pressure?

This picture of the diamond structure was made by Greg Egan back when we were working on 'topological crystals'.  The De Beers quote is from here:


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The icosahedron that got away

Iron pyrite can form crystals shaped like icosahedra.  They aren't regular solids, with equilateral triangles as faces - that would violate the laws of math!  Iron pyrite is a cubical crystal, and you can't make a regular icosahedron using little cubes.

These crystals are called pseudoicosahedra.  They take advantage of how the golden ratio can be approximated using Fibonacci numbers:

1/1 = 1
3/2 = 1.5
5/3 = 1.6666...
8/5 = 1.6125

and so on, getting closer to

Φ = 1.6180339....

They call iron pyrite fool's gold - and it can fool you into thinking its proportions attain the golden ratio.

Recently the curator of the Museum of Evolution, Palaeontology and Mineralogy in Uppsala, Sweden, emailed me and told me that the handsome pseudoicosahedron shown here was on sale for just $30.  I am very lucky to have friends like this.  His name is Johan Kjellman.

Unfortunately, I waited a day or two.   By the time I offered to buy it, it was already sold.   :-(

I wanted to buy it from here:

It's from the Merelani Hills near Arusha in Tanzania.  It was 2.8 x 1.4 x 1.3 centimeters in size:

A crystal of pyrite is set in calcite and graphite matrix. This find at Merelani has produced some of the most remarkable crystals in terms of crystallography. In this case the icosahedron! The luster and form are unmistakable from this locality. The back side of the crystal has damage but the display face is fine.

If you ever see a pyrite pseudoicosahedron for sale, let me know!  I'd also be happy to have a pyritohedron, which is nature's attempt to create a regular dodecahedron using little cubes.

For the math of the pseudoicosahedron and pyritohedron, go here:

and scroll down until you start seeing pictures that look right.


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Poncelet's Porism

If you can fit a triangle snugly between two circles, you can always slide the triangle around.  The triangle may have to change shape, but it stays snug!   All 3 corners keep touching the outside circle, and all 3 sides keep touching the inside circle.

That's really cool.  But even better, it also works for polygons with more than 3 sides!

This amazing fact is called Poncelet's Porism

A porism is like a theorem, but much cooler.  Poncelet was a French engineer and mathematician who wrote a famous book on 'projective geometry' in 1822. 

What's a porism, really? 

Well, Euclid is famous for his Elements, but he also wrote a more advanced book called Porisms.  Unfortunately that book is lost.  I hear that someone checked it out from the library of Alexandria and never returned it.   By now the overdue fee exceeds the annual GDP of Greece, so we'll never see that book again... and we'll never know exactly what Euclid meant by 'porism'.

Wikipedia starts by saying:

A porism is a mathematical proposition or corollary. In particular, the term porism has been used to refer to a direct result of a proof, analogous to how a corollary refers to a direct result of a theorem. In modern usage, a porism is a relation that holds for an infinite range of values but only if a certain condition is assumed, for example Steiner's porism.  [...]  Note that a proposition may not have been proven, so a porism may not be a theorem, or for that matter, it may not be true.

In short: nobody knows what a porism is, but people are willing to make stuff up.

Pappus of Alexandria managed to write down a few of Euclid's porisms around 400 AD, before the book got lost.  They are quite advanced facts about geometry.  Poncelet was inspired by Pappus, so when he proved his cool result, maybe he wanted to call it a porism too.  I don't know.

A slick modern proof of Poncelet's porism uses 'elliptic curves'.  Check out David Speyer's explanation:

In case you're not a mathematician, beware!  An 'elliptic curve' is jargon for a surface shaped like a doughnut.  We just call them 'elliptic curves' to keep people like you confused.

For some truly amazing connections between Poncelet's Porism and other math problems, see this paper by J. L. King:

An elementary proof of Poncelet's Porism is here:

In math, 'elementary' means that we don't use fancy concepts.  It doesn't mean 'easy'. 

When I retire, I want to quit proving theorems, and prove a porism.

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