The dual {5,3,4} and {4,3,5}

I've been collaborating with +Henry Segerman to create some physical models of hyperbolic honeycombs, and here is one of the first we made, the {5,3,4}.  If you would like to order one of these for yourself, information is at the bottom of this post.

What is a honeycomb?  

It is when you take a bunch of polyhedra and pack them together with no gaps.  The polyhedra in a honeycomb are called "cells".  This particular honeycomb is a perfectly regular repeating pattern of dodecahedra, four surrounding each edge.  For this reason, it is also called the Order-4 Dodecahedral Honeycomb.

Why is it hyperbolic?  

Hyperbolic geometry is a space where the angles of a triangle add up to less than 180 degrees.  The other kinds of geometries are Euclidean (angles add to 180 degrees) and spherical (angles add to more than 180 degrees).  If you try to pack four dodecahedra around an edge in Euclidean or spherical space, they don't fit because angles are too big.  But in hyperbolic geometry, you can make them fit.

Hyperbolic space can't fit into our normal Euclidean space, so this is a model of hyperbolic space called the Poincaré Ball model.  The infinity of hyperbolic space is compressed to the size of a ball, and as you might imagine, that results in some distortion.  Dodecahedra appear smaller and smaller as they approach the spherical surface of the ball, but the dodecahedra are all identical in hyperbolic space.  Although visually warped, the ball model preserves all angles: it is "conformal".

Our prints render edges of the honeycomb, and we've scaled them accurately (verses making them constant width).  For this reason, the edges appear thinner towards the ball surface.  The edge shrinking dictates how much of the honeycomb we can include, because as you add more layers of cells, edges eventually become so thin that a printed model would be too fragile.

What does the cryptic "{5,3,4}" mean?  

That is something called the Schläfli symbol, and it is wonderfully descriptive (and recursive!).  We can start at the left and work our way to the right to understand the meaning and build up a honeycomb.  The Schläfli symbol for a pentagon is {5} because it has 5 sides.  The symbol for a dodecahedron is {5,3}, which means that 3 pentagons meet at every vertex.  So the first number specifies a particular polygon and the second number describes how to connect up those polygons around a vertex.

The third number continues the trend.  It describes how to connect up cells specified by the first two numbers.  So the 4 at the end of {5,3,4} means that four dodecahedra surround each edge.  Each time we add a number to a Schläfli symbol, we end up describing an object of one higher dimension: first polygons, then polyehedra, then polytopes...

Something else is hidden in the Schläfli symbol of our honeycomb: the last two numbers taken together have meaning.  For the {5,3,4}, we're talking about {3,4}.  Remember {3} represents a triangle, and {3,4} means that four triangles meet at every vertex.  This is the octahedron.

How is the octahedron is meaningful for this honeycomb?  It is the the vertex figure, which describes the geometry of how dodecahedra meet at each vertex in the honeycomb.  Eight dodecahedra meet at every vertex, one for each face of the octahedron.

What a crazy amount of information compressed into three numbers!

There are four regular honeycombs in hyperbolic space (and eleven more if you loosen the definition a little).  Each of these has a unique Schläfli symbol, and so the symbol is a great way to quickly understand how a particular honeycomb is built up.  The symbols also work for honeycombs in Euclidean and spherical space.

Now let's check out the {4,3,5}

Can you break down the symbol to understand how this one is built up?  What polygon lives throughout?  What are the polyhedral cells?  How many cells fit around an edge?  What is the vertex figure?

The {4,3,5} is the "dual" of the {5,3,4}, so locations of all the element types are exchanged.  Where there are cells, polygons, edges, and vertices in one, you respectively have vertices, edges, polygons, and cells in the other.  And maybe you just noticed something else about the Schläfli symbol... dual objects have reversed symbols.  Self-dual objects have palindromic Schläfli symbols!

You can order your own prints of both these models (and more exotic honeycombs) at my shapeways shop.

        http://www.shapeways.com/shops/roice3

If you'd like to dig more, see Coxeter's paper, "Regular Honeycombs in Hyperbolic Space", available in Chapter 10 of the book "The Beauty of Geometry" or online here.

Three Different Views of the Same Tiling

Here are three images of the same {4,6} tiling. The "{4,6}" notation is called the Schläfli symbol. The 4 means it is a tiling of squares. The 6 means that 6 squares meet at every vertex. Check it in the pictures!

A view centered on a square:

A view centered on an edge:

A view centered on a vertex:

I bet you noticed that the tiling is checkerboarded (hyperbolic chess anyone?). Can you figure out what property is required to allow a tiling to be checkerboarded? If so, let me know in the comments!

Q: What made these pictures?

A: The MagicTile v2 Preview (requires .NET 4 Framework).

Geodesic Saddles

It's highly likely you've seen a geodesic dome before.

After briefly starting to optimize triangle counts for textures in MagicTile, I had a fun realization. The triangle patterns sparked the idea that there could be a precise hyperbolic analogue to a geodesic dome. I was compelled into the diversion, and with minor code changes made some pretty pictures of "geodesic saddles". (That seems like a nice name for these objects anyway.) Alas, my intended optimizations are yet to be done, but at least I can present this geodesic saddle based on the {3,7} tiling :)

Can you find some of the "knots"? That may not be the proper term, but I mean those rare points in the saddle where seven triangles meet at a vertex instead of six. On a geodesic dome, which is usually based on the spherical {3,5} tiling (aka icosahedron), the analogous points are the rare vertices where five triangles meet instead of six. I find knots easier to spot on a geodesic saddle derived from a {3,9} tiling.

Geodesic domes and saddles are generated by taking the tiles in a triangular tiling and subdividing each of them into smaller triangles. Hence, triangular numbers make a cameo in the calculations. For these pictures, I chose to subdivide the original triangles with eight new triangles per side.

But there was one thing that tripped me up quite a bit. I began by mistakenly thinking I could subdivide the triangle edges of the original tiling equally, and then interpolate interior points thereafter. As much as I tried, things just wouldn't line up quite right, and I wasn't seeing the geodesics that I expected. It turns out that all the small triangle edges have varying lengths, something that is also true for a geodesic dome. Compare the proper {3,9} geodesic saddle above with the waviness of an incorrect effort.

I've been showing these pictures in the Poincare Disk, and I don't have unprojected renderings at the moment. But despite this, one thing is certain - a portion of geodesic saddle would make for a unique and fantastic jungle gym!

Related links:

Hexagons, Pentagons, and Geodesic Domes

Some similar pictures I found after my experiments

Happy Bugs

Programming mistakes might usually lead to a crash, lost work, tears, etc., but here are a couple pleasant surprises courtesy of MagicTile coding goofs.

The following was caused by a coloring bug. I accidentally left out a line of code, which caused the yellow and black outline colors to take over in a completely unexpected way.

This past weekend, I was testing slicing off the tips of polygons in a hyperbolic tiling, and saw this beautiful, fortuitous image appear on my screen.

The cause... I had failed to tell the slicing code to recalculate the centers of the sliced up pieces, so they were way off from where they should have been. The intended (much less satisfying) outcome was this.

I liked the unanticipated effect, so captured a couple more pictures before fixing the bug :)

Knitting Math

In the January issue of the Notices of the American Mathematical Society, I fortuitously saw an A K Peters publishing ad for a new book by Daina Taimina titled Crocheting Adventures with Hyperbolic Planes. A whole book by an author of the paper I had previously found!  It wasn't then available, but I preordered a copy immediately, and today I excitedly received it.  This is a beautiful book, and I'm going to love it.  After flipping through and enjoying the multitude of pictures, the forward by William Thurston started off the reading experience perfectly.

Many people have an impression, based on years of schooling, that mathematics is an austere and formal subject concerned with complicated and ultimately confusing rules for the manipulation of numbers, symbols, and equations, rather like the preparation of a complicated income tax return, where there are myriad unexplained steps, rules, exceptions, and gotchas.
        Good mathematics is quite opposite to this. Mathematics is an art of human understanding.

My first impression is that this is a gentle, real-world-examples and hands on introduction to hyperbolic geometry.  My sneak peek also showed there's more crocheting projects for me to learn in Taimina's work, and I'll surely be breaking out my hook again.  I have to say I'm liking having this bridge between Sarah's and my interests.  

By the way, "Knitting Math" was how a friend of ours described my crocheting adventures to some others when the term hyperbolic plane slipped her mind. I thought it was great :)

Reno and Mathematical Earrings for the Holidays

Sarah and I went to Reno over the holidays with her brother Jon and her parents Larry and Paulette. We were visiting her grandparents Fred and Bev, who live about 40 minutes outside Reno in a beautiful, isolated area. Cell phone reception was a fraction of a bar, and this was a plus for getting away. The "kids" (I'll never grow up) got to camp out in their trailer, which turned into a big adventure on Christmas night with the heater broken and freezing temperatures. Jon, Sarah, and I sequestered ourselves into the small bedroom with a surface heater, and it was fun - probably much easier for Sarah and me since we could cuddle up. There was no snow when arriving or leaving, but we had a lucky white Christmas! We also snuck in two half-days of snowboarding, the first time ever for Jon, Sarah's second outing, and my first in about two years. That was more great fun, and here are some pics and video from the week. Sadly, when taking pictures in the snow, I'm like Chandler in that one Friends episode.

On the return home, Sarah saw a pair of earrings in the airport that she thought I would like for mathematical reasons. She was right on! Any time circles get involved, you can pretty much bet things have an interesting mathematical interpretation, but these earrings are considerably rich for generating discussion.  And lucky for Sarah, her mom bought them for her.

If we consider the outer circle as the boundary of the Poincare disk model of the hyperbolic plane, there are three horocycles and one h-circle in this earring. The horocycles are tangent to the boundary, or "circle at infinity", and the h-circle is floating in the interior. By h-circle, I mean residents of the hyperbolic plane would see it as a circle, that is a set of points a fixed distance from a center. The Euclidean center of the h-circle in the model (what you see as the circle center) doesn't coincide with the center of the circle the residents would perceive however - that only happens for h-circles having centers at the Euclidean center of the disk.

No geodesics (straight h-lines) are in these earrings, as those would be segments of circles orthogonal to the boundary of the disk. Two of the horocycles (one is the tiny circle with the stone in it) are tangent to each other, meeting at a single point. Noticing this reinforced their non-geodesic character to me, for if they were straight lines and met at some interior point, they'd necessarily intersect.

Moving away from hyperbolic geometry, it is also neat to note the three intersecting circles in the middle are mutually orthogonal to each other! (or quite close to it anyway given the imperfections of this physical model). What's cool about that is inverting any of those three circles in the others will leave the shape invariant. Furthermore, if we invert any two of the circles in the third, the two intersection points of the first two circles are swapped.

While not as directly related, the earrings are reminiscent of a Wolfram Demonstration recently posted by Ed Pegg, Jr. about The Circles of Descartes. On that page, he shares a portion of a poem by Frederick Soddy called The Kiss Precise. I've also seen this poem in an engaging biography of the geometer Donald Coxeter called King of Infinite Space. The theorem described by the poem isn't exhibited in the earrings (close, but there are only three mutually tangent circles, not four), but as I still feel it is befitting, here is the full version (I like the generalization to spheres in the third verse).

The Kiss Precise by Frederick Soddy

For pairs of lips to kiss maybe
Involves no trigonometry.
This not so when four circles kiss
Each one the other three.
To bring this off the four must be
As three in one or one in three.
If one in three, beyond a doubt
Each gets three kisses from without.
If three in one, then is that one
Thrice kissed internally.

Four circles to the kissing come.
The smaller are the benter.
The bend is just the inverse of
The distance form the center.
Though their intrigue left Euclid dumb
There's now no need for rule of thumb.
Since zero bend's a dead straight line
And concave bends have minus sign,
The sum of the squares of all four bends
Is half the square of their sum.

To spy out spherical affairs
An oscular surveyor
Might find the task laborious,
The sphere is much the gayer,
And now besides the pair of pairs
A fifth sphere in the kissing shares.
Yet, signs and zero as before,
For each to kiss the other four
The square of the sum of all five bends
Is thrice the sum of their squares.

        - Nature 137, 1021 - 1021 (20 Jun 1936)

Hi, my name is Roice... and I crochet

Picture me and Sarah sitting side by side in bed around 11:30 pm, her knitting, me crocheting, and you'll have a pretty good idea of a number of evenings of ours over the last few weeks.  Maybe not the most exciting image, but we've been having fun.  I've finished my first crochet project now, a portion of a hyperbolic plane with radius of curvature of about 5 cm, and I think it turned out pretty good.  The crocheted rows are quite visible in this picture I snapped.  Are they geodesics of the surface? *

I'm sure I'll learn more by playing with it, but I've learned some already, not the least of which is that I can't count to 5.  All I had to do was 5 normal stitches for every doubled-up stitch, and I'd say 60% of the time I lost my place!  And boy are programmers spoiled with undo.  Too bad that functionality isn't available in the physical universe.

Something noteworthy about this particular construction (but not a property of hyperbolic geometry itself) has to do with the fact that the number of stitches in successive rows forms a geometric sequence, that is the length of each row is a constant multiple of the previous row.  That has some unintuitive side effects.  I did 23 rows total, the first had 20 stitches and took maybe a minute, but the last had over 1000 stitches and took almost 3 hours!  If I were to do another 23 rows, the final 46th row would take me over 28 days (no sleep, no breaks) and who knows how many skeins of yarn.  Add yet another 23, and the final row would take over 5 years.  This reminds me of "the magic of compounding interest", and what I've been told the value stocks are supposed to do in theory.

Speaking of economic unraveling, this leads to something else intriguing about my hyperbolic plane.  Instead of tying off the end when I was done, I could have undone the entire uber-knot in one fell swoop just by pulling out my crochet hook and gently pulling on the yarn.  It's like the whole thing is a house of cards, a deceivingly stable form that is actually no more substantial than the first slip knot that started the whole thing.  This reminds me of the axiomatic foundations of mathematics.

While working on this, I couldn't help but focus on a possible useful application.  I haven't figured it out yet, but my mind can't let go of the idea that this could solve the widespread problem of competition for blankets when couples sleep.  The extra material seems like a perfect candidate to provide some benefit here :)

* Nope.  If they were, I would be able to fold the surface so that they appeared flat and straight.

Sarah Goes Hyperbolic

Sarah has been knitting some pretty scarfs lately. She was showing off her latest project to me, and lo and behold it turned out to be mathematical! As she was knitting successive rows, she added incremental stitches to give it a ruffled appearance. I told her I thought this was especially cool because the extra stitches were giving the scarf a negative curvature. It was hyperbolic! That is what happens when you try to put extra material into what would otherwise be a flat 2 dimensional surface.  She lovingly rolled her eyes :)

Every time I think I have a new idea, it turns out someone has already been there, done that.  On the plus side, the article I then tracked down already contained developed information and instructions for crocheting your very own hyperbolic plane.  Following the directions will result in a hyperbolic surface of constant negative curvature (Sarah's scarfs don't adhere to the constant part).  I also found this site with some nice pictures of completed crochetings (the site mentions the work of Daina Taimina, who is one of the authors of the paper above).

It is interesting to note that if you build a constant negative curvature surface large enough, it will necessarily end up intersecting itself in our 3D world.  Models living in our physical universe are limited in their representation.  This is in contrast to models of constant positive curvature surfaces, which do fit nicely into the world.  The surface of any ball will do.

Sarah and I just returned from Hill Country Weavers, where Sarah bought me a crochet hook, so I'm now off to attempt creating my own hyperbolic plane!

update:  Sarah did not aprove the cuteness factor of my first picture, so I've uploaded an improved version.

Hyperbolic Tiling in Motion

I've seen a number of pretty images of hyperbolic tessellations (tilings) on the web. A nice gallery from one of the producers of the Dimensions videos turns many flat Escher tilings into hyperbolic ones. What I hadn't seen until I went looking the other day was a way to animate motions of the hyperbolic plane (using the Poincare Disk model).

I thought this would be a cool project, and decided to do a quick search to see what applets out there might already be doing it. I was pleased to find this one right at the top of the search results (and even more pleased because I know the author Don Hatch through the Rubik hypercube group and was able to meet him some time ago...wow actually that was almost a decade ago...scary).

Drag your mouse around on his applet above to see this tiling of the hyperbolic plane translate (update: Sarah let me know this doesn't work in Google Reader btw). This can give a much better feel than a static picture that each tile in the image is actually the same shape (a regular polygon). I know this post is lacking in background, but hyperbolic geometry can't be represented through normal plane geometry without distortion, hence there are alternate representations with tradeoffs in characteristics. One thing to notice is that the white tiles have 7 sides and the dual blue tiling has 7 triangles meeting around a point, but you can't have such a tiling in normal space with a set of regular triangles (7*60>360). There just isn't enough space, kinda like our closet storage for Sarah's shoes :)

This applet is highly configurable and fun to play with, e.g. click on it and start pressing p. You can find all the info on Don's applet page. There are some extensions I would still like to see in addition to translations. A rotation of the plane about any selected point would be cool. Hyperbolic geometry is more interesting than flat (Euclidean) geometry, leading to more possibilities as well. Specifically there is a special kind of rotation called a limit rotation, and this would also be neat to see. Finally, it would be sweet to allow animated motions of other models of hyperbolic geometry like the half plane model. So there is still fun potential hobby coding to be done (of course).