Below is a modular origami construction known as the 120-unit sonobe ball. It has the geometry of an icosidodecahedron but with the 12 pentagonal faces of the icosidodecahedron split into five of the little equilateral pyramids that sonobe units demand.

The above was made from 120 Post-it notes, three sets of forty in each color. I haven’t seen anyone post a picture of one of these 3-colored like this, i.e. in three colors such that no two units of the same color are adjacent to each other, so I thought it was of some interest…

A schematic of the coloring I found is below. I have no idea if it is unique:

I’ve decided my “untitled Escher game“, i.e. Zoop on a double logarithmic spiral, is going to be named “Draak”, which is the Dutch word for dragon. The title bears no relation to the game beyond the fact that I am going to use as its logo art based on the following novel tesselation of the plane that I discovered while fooling around with the tesselation tool that I implemented to construct the sprites for this game, and also that I thought it would be cool to make the title art be an ambigram with rotational symmetry and I think I could figure out a way to do this with the word “draak”.

Over the past several weeks I have been investigating desktop applications for creating novel Escher-like tilings of the plane. Basically I’ve determined that none of what is publicly available is useful to me. The game I have in development will involve an animated Escher-like tessellation of a double logarithmic spiral — see here. The “logarithmic spiral” part of it means the publicly available applications can’t help me: they are too limited in what they will do and I can’t extend them because they are proprietary, so I have created my own tool.

These applications are “too limited” in the sense that none encompass everything possible that one would regard as a 2D tessellation. You have some applications that simplistically shoehorn you into a few kinds of geometries ignoring the richness of the whole domain of tiled patterns. You have other applications that try to take the domain seriously and follow the mathematical literature on the subject. What this means, generally, is that they allow the user to construct a tessellation in terms of its “wallpaper group” which is the mathematical classification of a plane pattern based on the symmetries the pattern admits.

The latter seems like what one would want but in practice it isn’t, at least it is not what I want. It is too heavy of a constraint to work only in terms of wallpaper group because there are all kinds of patterns one might be interested in that fall outside of this formalism: Escher’s tessellations of the Poincare disk model of the hyperbolic plane, for example, or, say, any kind of aperiodic tiling or any tessellations that admit similarity as a symmetry, i.e. scaling, such as Escher’s square limit etchings, or my case: various spiral patterns.

It is instructive that Escher himself did not work in terms of symmetry groups. He created his own classification system which Doris Schattschneider refers to as Escher’s “layman’s theory” of tilings in her book Visions of Symmetry (the authoritative text on Escher’s tessellations). Escher’s classification system comes from a pragmatic perspective rather than a formal mathematical perspective. He starts with basic cells — a square, a rectangle, an isosceles right triangle, etc. — and then enumerates the ways in which one can surround the cell with itself that result in tilings of the plane. There is a good overview of his system here.

What I like about Escher’s approach is that it is geared towards construction. When you try to make one of these things you start with a tile or some a small set of simple tiles that tile the plane. You try to modify them until you get something you like. I decided to implement software that captures that approach. I didn’t want to have to be too concerned with “wallpaper group” or other abstractions which I don’t personally find to be intuitive anyway.

What I came up with is the following application that is written in C#. Below is video in which I use the application to construct something like Escher’s butterfly tessellation — system IX-D in his layman’s theory — but on a double logarithmic spiral (This is going to be the basis for the art of the first level of my game; the butterflies will be animated, flapping their wings, etc.)

The way the application works is that there are two kinds of tiles: normal tiles and tile references. Normal tiles are composed of “sides”. There are two kinds of sides: normal sides and side references. Tile references and side references point to a another tile or side and say basically, “I am just like that tile or side, except apply this transformation to me”. The transformations that tile references can apply to a tile are defined in terms of affine transformation matrices. The transformations that side references can apply are just two kinds of flipping relative to the sides’ end points, either flipping “horizontally” or flipping “vertically” or both. The application then allows the user to add and move vertices on any sides that are on tiles that are marked as editable and then resolves all the references in real-time.

Right now, all of the information about tiles and sides and which is a reference to which has to be hand coded as an XML file, which is a pain (for the above I wrote a separate Python program that generated the XML defining a double logarithmic spiral of tiles that interact with each other according to Escher’s IX-D system), but it is an extremely flexible formulation (it could, for example, handle hyperbolic tessellations in theory … if you added a kind of side that is a circular arc and moebius transformations to the set of transformations the applications knows how to apply to referenced tiles).

Eventually I’d like to release this application as open source but I am not sure it will be useful to anyone but me in its current form. I need to incorporate the part that one has to hand code in XML into the actual GUI … fix bugs and bezier curves and so forth, but please feel free to contact me if you are interested in the application in its current state or otherwise.