( Progress on the ) tiling printer.

Uni-color-2D graphics can't be spectaculair, let alone breathtaking. Unless you are in the know. Let me explain. I am now able to - almost - print tilings automatically from just a string of numbers as I wrote about in a previous post.

Not spectaculair at all, it's just a collection of 8 tilings containing 72 polygons in total. But if I change ( 4,2 ) by (10,5) and add a colorfunction the program creates immediately the following graphic.

Both are, in fact, representations of the same sequence.{6, 2, 4, 3, 3, 2, 4, 2, 3, 2, 3, 3, 4, 4, 3, 3, 4} but the variations from this sequence alone are endless. The program creates a graphic for every sequence input. Not all graphics will be tilings of the plane by definition. The following ( major ) step is to let the program reject sequences that do not lead to tilings of the plane.

Tne next part will include an implentation of a 'Point in Polygon' predicate. Many of those exist and have been implemented for many languages. That could be a topic for a more in-depth post next time.

Some progress...

If you read the previous posts on my tiling printing algorithms you'll understand the reason for this post: I made some progress that unraveled some serious knots in my stomach.

{ 4, 2, 4 } ->

{ 6, 2, 4, 3, 3, 2, 4, 2, 3, 2, 3, 3, 4, 4, 3, 3, 4 } ->

The second white polygon is made from the following points :
\begin{array}{cc}
-\frac{\sqrt{3}}{2} & \frac{1}{2} \\
-\frac{\sqrt{3}}{2} & -\frac{1}{2} \\
0 & -1 \\
\frac{\sqrt{3}}{2} & -\frac{1}{2} \\
\frac{\sqrt{3}}{2} & \frac{1}{2} \\
\frac{1}{2} \left(1+\sqrt{3}\right) & \frac{1}{2} \left(1+\sqrt{3}\right) \\
\frac{1}{2} \left(1+\sqrt{3}\right) & \frac{1}{2} \left(3+\sqrt{3}\right) \\
\frac{\sqrt{3}}{2} & \frac{3}{2}+\sqrt{3} \\
0 & 1+\sqrt{3} \\
-\frac{\sqrt{3}}{2} & \frac{3}{2}+\sqrt{3} \\
\frac{1}{2} \left(-1-\sqrt{3}\right) & \frac{1}{2} \left(3+\sqrt{3}\right) \\
\frac{1}{2} \left(-1-\sqrt{3}\right) & \frac{1}{2} \left(1+\sqrt{3}\right) \\
\end{array}

Ready to enter the next level of the problem. ;-)

Closed PolyLine - (2)

I developed an algorithm that takes sequences like $\{ 6, 2, 4, 3, 3, 2, 4, 2, 3, 2, 3, 3, 4, 4, 3, 3, 4 \}$ and turns them into drawings like this ( in Mathematica, of course ).

In order to determine if the tile above tiles the plane I need to remove all internal lines...

... which turned out a tad more dificult than expected. I got as far as a prototype algorithm which I am currently testing and improving.

To be continued.

Closed PolyLine

The picture ( below) contains two drawings ( created with Mathematica ), the drawing on the left consists of a hexagon, a square, a triangle and again a square. In order to facilitate an algorithm that decides if this drawing tiles the plane I need a closed polyline to determine if a point is within the borders of the drawing. Oddly enough it is more difficult to create the drawing on the right than the original on the left.

The drawing on the right is a closed polyline, it consists of a number (11) of line segments and has no begin- and endpoint, it is closed. ( In another project I am working on we call PolyLines, MultiLines but I haven't seen that name in use elsewhere. )

Geometric cell division

Take a square and add a similar square to one of its sides and remove the shared edge. What remains is a rectangle.

Take an octagon and add a similar octagon to one of its sides and remove the shared edge. What remains is the following polygon.

 Take an n-gon and add a similar n-gon to one of its sides and remove the shared edge. At infinity the n-gons will divide into two circles. Unfortunately my computer is too slow to effectively record this in a video.

n=12

n=32

n=64

Example of a polygonal tiling

The tiling in the image below is edge-to-edge, polygonal, but not uniform because it has different vertex types, i.e. (3,3,4,3,4), (3,4,4,6), (3,4,6,4) and (3,6,4,4). It is therefore NOT an Archimedean tiling,

With the following piece we can tile the entire plane...

... as we can see here.

Printing polygons edge-to-edge

Recipe
Suppose you have some polygon with cornerpoints { p1, p2, ..., pk } and you want to print a regular polygon with n edges along one of its edges (p(j), p(j+1) then you can simply find the first point of the regular n-gon by rotating p(j+1) with centre p(j) over 360/n degrees. You can continue this process until you have found all points or you can calculate the centre of the regular polygon, its orientation and edge-length which you need to print a regular n-gon. Here are some examples.

Examples

3-on-4, 4-on-3 and and 5,7,9,11-on 4 (dark-on-light ).

The (3,12,12) Tiling

I made ( what programmers would call ) a 'recipe' for creating a program that prints an Archimedean Tiling. Needless to say that I am talking about Mathematica code. I basically have a program that can distribute any set of Mathematica Graphics objects over a lattice of points. So the recipe basically means calculating the points of the motif, putting them in a set and handing them over to the printer. The last one I did is (3,12,12) Tiling.

Tiling with vertex type (3,12,12).

The next step is to abstract and code the recipe itself, i.e. translating lists like (3,12,12) or (3,3,4,3,4) to graphics. In fact, I calculated the points for (3,12,12) with a first version of that program. -

Johannes Kepler and (3,3,4,3,4)

Who doesn't know the name of Johannes Kepler? Kepler (1571 - 1630 ) formulated the laws of planetary motion and his work provided the foundation for Isaac Newton's theory of gravity. My point being that Kepler was a scientific giant in his days and his name will live on forever.

My current mathematical project ( personal challenge if you like ) is focused towards tilings of the plane, creating ( Mathematica ) software to print and generate tilings. And ultimately -find- new tilings I haven't seen before. There aren't many textbooks on the subject, the classic work is very recent ( in mathematical terms ), it was published in 1986: Tilings and Patterns, by Branko Grunbaum and Geoffrey C. Shephard

Earlier today I was doodling on the tiling (3,3,4,3,4) of which I uploaded a picture.

Doodle of (3,3,4,3,4).

Then, to my surprise I read in Grunbaum / Shephard that it was Johannes Kepler (!) who started the mathematical research on tilings and patterns. One of the beautiful books Kepler wrote is called the Harmony of the World, originally published in 1619 but recently translated into English by Aiton, Duncan and Field and published the American Philosophical Society.

The Harmony of the World consists of 5 books.
- Book 1: On the construction of regular figures
- Book 2: On the congruence of regular figures
- Book 3: On the origins of the harmonic proportions, and on the nature and differences of those things which are concerned with melody
- Book 4: Preamble and explanation of the order
- Book 5: ( No title ).

This is part of a picture ( drawing ) from book 2 which has a drawing of (3, 3, 4, 3, 4 )  just like my doodle rype  marked as O ( top right ).

This proves once more that mathematics, by itself, does not change over time, its timeless. At least this part ( if not all ) of mathematics has to be discovered. The tilings of the plane have always been there, it just takes us to see them so that we can ultimately categorize them.

The fact that Kepler worked on this makes him human ( but still a giant of course ) to me, I can imagine the joy and excitement he must have felt drawing the illustrations especially since they take a lot of ( behind the scenes ) calculations.

Archimedean (4,6,12) tiling - Color patterns

Once you programmed the printing of a tiling pattern it is very easy to add colors to the tiles. Some examples.

Archimedean (4,6,12) tiling

This is ( part of ) the Archimedean (4,6,12) tiling.

Archimedean (3,4,6,4) tiling - In color

Tilings become appealing when they are colored.

Archimedean (3,4,6,4) tiling

This is ( part of ) the Archimedean (3,4,6,4) tiling.

The (3,4,6,4) means that at every vertex you'll find four tiles with 3,4,6 and 4 vertices respectively. The Archimedean tilings are vertex-uniform.

Visual Mathematics

Years ago when I started studying mathematics besides my job in IT I never considered that I would be able to apply mathematics in my day-to-day job. But I do now, because I work on the development of a drawing program ( a specialized drawing program on Android ). Does that make me happy? Just a bit. Because  applied mathematics can get as dirty as computer programming. Once applied, mathematics has lost most if not all of its beauty ( although nothing of its power ). I love pure mathematics, and even more so visual mathematics. The three pictures below are the result of applying a function to respectively the integers 1, 2 and three. The construction ( Mathematica programming, if you like ) of that function however required understanding of calculus, geometry, linear algebra, group theory and tling theory ( they are all parts of the Archimedean tiling of the whole plane with vertex type 4,8,8 ). My point being: this is the mathematics I like so much.

When I really like a picture I made I add it to 'The Gallery'. I am far from being able to create art with mathematics but there is a point where mathematics becomes art or where art becomes ( laying the groundwork for future ) mathematics. M.C. Escher explored mathematics decades before general theories about the subject were formulated.

If you are interested in Mathematics and Art then I can recommend this book: Connections: The geometric bridge between art and science.

Tilings resources

I am on a OU course again, ( more about that in posts to follow ). The course site opened today, that really kicks off the course for me.

For now some resources you might find interesting.

A tiling is a covering of the whole plane with non-overlapping tiles, each of which is a topological disc. The classic work on tilings is Tilings and Patterns by Grunbaum, Shephard. A list with other books on the subject can be found here.

M.C. Escher used tilings in his graphics work in an ingenious way. This book contains a nice collection of the work of Escher.

It's interesting to note that in Escher's time (1901-1972) there was hardly any mathematical theory about tilings. The foundational work on tilings was published five years after Escher died. Yet from Escher's work it is clear that he understood tilings, and the related line symmetries ( Frieze Patterns ), lattices and plane symmetries ( Wallpaper Patterns ) as no other.

If you are interested in puzzles at all it's likely that you came across Jaap's Puzzle Page, a vast resource of information regarding puzzles. The website is maintained by Jaap Scherphuis. You'll find his YouTube site here with many puzzle demonstrations.

To my astonishment Scherphuizen also maintains an impressive collection of tilings on a Tilings Page. His tilings demonstrations Java Applet is as impressive which you'll find on the same page.

'Pentagon Flower' ( background tiling is the [4,8,8] Laves tiling )
(c) nilo de roock 2012