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.

Regular polygons, exercise.

With a drawing program yesterday's pictures are easy to fake, of course. But these drawing programs don't give you the numbers.

Exercise.

Place a decagon edge-to-edge on a square with sides of length 1 ( see figure ). What is the distance between the two marked points?

( Answer: $ \frac{1}{4} \left(3 \sqrt{10-2 \sqrt{5}}+\sqrt{50-10 \sqrt{5}}+4\right) $ )

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.

OU exams more difficult than ever ( ... ) ?

I read a rumor on facebook that the Open University exams were harder than ever. No numbers were shown to substantiate the claim however. You may have been aware that the OU rates have been increased dramatically to align them with the rates of the "Brick Unis" ( = how normal universities are called in OU jargon ). Now one of the commenters said that they are doing the same thing with the exams. Suggesting that until now OU exams were much easier than Brick Uni exams. - To be honest I think it's the other way around. Often homework assignments ( for maths at least ) are part of the grade Brick Uni exams while at the OU you get the lowest grade of homework and exam.

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.