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

Tragedy of the Commons (and more)

The whole world is a comedy to those that think, a tragedy to those that feel.

-Horace Walpole

Can you guess which camp I'm in? My answer told me a little something about me.

I tip my hat to AbstruseGoose for the quote (my favorite online comic by orders of magnitude)...

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 :)

Appreciating Creativity

I've always liked reading the dedication in books before I start them. Here are a couple I ran into this past year whose creativity made me smile.

from On Quaternions and Octonions by John Conway and Derek Smith:

This book is dedicated to Lilian Smith and Gareth Conway, without whom we would have finished this book much sooner.

from Nonplussed!: Mathematical Proof of Implausible Ideas by Julian Havil:

To Anne
for whom my love is monotone increasing
and unbounded above

I can relate to both sentiments :)

Happy Holidays

What is this?

Besides possibly a festive tree ornament, it is a sneak preview of a new Rubik analogue program I've been playing with this past year but have yet to publish (update: I put a version of this online in late January). One can bring to light a huge set of puzzles by considering the original cube puzzle as a special case where the colored faces are a regular tiling of squares, and then abstracting by asking "What is the most precise analogue for other polygonal tilings?". For me, this simple question lead to all kinds of engrossing pathways, discoveries, and of course... many more questions.

We live on an island surrounded by a sea of ignorance. As our island of knowledge grows, so does the shore of our ignorance.

- John Wheeler

I saw that quote this week on John Baez's site, and ground zero on my personal island appears to be the Rubik's Cube.

Anyway, the picture above is a checkerboard pattern made by twisting up a puzzle based on a regular tiling of octagons, which requires hyperbolic geometry to fit together, naturally! The octagonal faces are delineated by yellow edges, and the black circles slice them up into stickers. This particular puzzle has 6 unique face colors which repeat in a certain pattern, and that pattern is interesting to study in itself. All the faces having the same center colors are actually identical and twist together. Kaleidoscopical to watch :D

Best wishes to all my readers in 2010. I love all three-or-so of you!

In Praise of (Blogging) Idleness

This entertaining essay by Bertrand Russell begins:

Like most of my generation, I was brought up on the saying: 'Satan finds some mischief for idle hands to do.' Being a highly virtuous child, I believed all that I was told, and acquired a conscience which has kept me working hard down to the present moment. But although my conscience has controlled my actions, my opinions have undergone a revolution. I think that there is far too much work done in the world, that immense harm is caused by the belief that work is virtuous, and that what needs to be preached in modern industrial countries is quite different from what always has been preached.

So why the dystopia of the world instead of Russell's four hour work days? Scott Aaronson provided a good explanation of why it has to be this way:

Why can’t everyone just agree to a family-friendly, 40-hour workweek? Because then anyone who chose to work a 90-hour week would clean our clocks.

...

Again and again, I’ve undergone the humbling experience of first lamenting how badly something sucks, then only much later having the crucial insight that its not sucking wouldn’t have been a Nash equilibrium.

And Parkinson's Law surely plays a role in the dynamics of it all:

Work expands so as to fill the time available for its completion.

Whew, well that's enough posting for me - back to another 6 months of leisurable, blog-free bliss...