Mathematics in action: tiling the plane

In the next post of our mathematics in action series, we look at tessellations of the plane. The most familiar of these are the three regular tilings, using tiles that are regular triangles, squares, or (as below) hexagons.

Photo: Claudine Rodriguez

The great Dutch artist M. C. Escher is famous for his distorted versions of such tilings, such as this tiling on the wall of a museum in Leeuwarden:

Photo: Bouwe Brouwer

Alternatively, the regular tilings can be extended by mixing different kinds of regular polygon. Of particular interest are the eight semiregular tilings, in which the tiles all meet edge-to-edge, and each vertex is equivalent to each other vertex (i.e. each vertex can be mapped to each other vertex through rotations, reflections, translations, or glide reflections). Here is one of the eight:

Photo: “AnnekeBart”

Because of the high level of symmetry, an exhaustive list of the 11 regular and semiregular tilings can be made by considering all possible meetings of polygons at a vertex, such as these two:


Penrose tilings, discovered by Roger Penrose in 1974, loosen the regularity and symmetry conditions, while still using a fixed number of kinds of tile, and while still being “almost” symmetrical. In the image below, Penrose is standing on a Penrose tiling. His 1974 discovery goes to show that fairly simple mathematical truths can still be discovered today.

Photo: “Solarflare100”

Mathematics in action: affine transformations and PostScript

Here’s the next in the mathematics in action series.

An important group of geometric transformations are the affine transformations, such as translations, reflections, and rotations. They are defined by a combination of a matrix multiplication and a translation:

Such transformations are an important part of the PostScript page description language. For example, the PostScript code below, after the initial header, specifies a black letter F, and another in blue, translated by (50, 50):

%%BoundingBox: 0 0 196 118
%%Pages: 0
%%DocumentFonts: Times-Roman
%%DocumentNeededFonts: Times-Roman
%%Orientation: Portrait
gsave newpath 100 dict begin /Times-Roman findfont 50 scalefont setfont

gsave 0 setgray 0 0 moveto (F) show grestore

gsave 0 0 1 setrgbcolor 50 50 translate 0 0 moveto (F) show grestore

The output is included in this picture:

But of course that is using only the (ef) part of the transformation. More interesting is a rotation through an angle θ, where the matrix is:

PostScript allows this to be specified using a translation and a rotation angle in degrees (red F) or as an (abcdef) matrix which combines the rotation matrix with a (180, 0) translation (grey F):

gsave 1 0 0 setrgbcolor 100 0 translate 30 rotate 0 0 moveto (F) show grestore

gsave 0.5 setgray 30 [0.8660 0.5 -0.5 0.8660 180 0] concat 0 0 moveto (F) show grestore

A diagonal matrix provides scaling in the x and y directions, with negative scaling factors giving a reflection. Again, PostScript allows the scaling factors to be provided directly (green F), or as an (abcdef) matrix (brown F):

gsave 0 0.7 0 setrgbcolor 100 100 translate -2 0.5 scale 0 0 moveto (F) show grestore

gsave 0.5 0.3 0 setrgbcolor [-2 0 0 0.5 175 75] concat 0 0 moveto (F) show grestore

end grestore showpage

Copying and pasting the three blocks of PostScript into a text file with a .EPS extension will give the coloured image above, which can be viewed by printing it, or by inserting it into a Microsoft Word document. Experimentation with other, more complex, affine transformations is easy that way.

Mathematics in action: dice

We are all familiar with dice of various kinds. With fair dice, each number will come up with equal probability, regardless of how the die is rolled. This requires a degree of symmetry – we want a die to be a polyhedron where all the faces are equivalent. The obvious candidates are therefore the five Platonic solids, in which not only the faces, but also the edges and vertices are equivalent. The Platonic solids give us the common d4, d6, d8, d12, and d20 dice:

Five Platonic dice (d4, d6, d8, d12, d20) and two pentagonal trapezohedra (d10) – photo by “Copat”

However, the Platonic solids are more symmetrical than necessary for the job. A tetragonal disphenoid, for example, makes a very good d4:

A tetragonal disphenoid makes an alternative d4 – photo by “Traitor”

What is required is that a die be isohedral (also called “face-transitive”). Each face should be equivalent. Specifically, for any numbers A and B, given the die with A on top, there should be a series of rotations and reflections that make the die look like the starting position, but with B on top. This rules out shapes like the gyroelongated square bipyramid, where all the faces are equilateral triangles, but the triangles are not equivalent (the “end” triangles differ from the “middle” triangles):

A gyroelongated square bipyramid does not make a fair die – photo by Andrew Kepert

We also want a die to be convex, so that it can land on its faces. Stellated polyhedra are excluded:

Stellated polyhedra cannot be dice – mosaic in St Mark’s Basilica, Venice

Trapezohedra satisfy this “convex and isohedral” rule, and the pentagonal trapezohedron is commonly used as a d10 (see the picture above). Trapezohedra work best with 10, 14, 18, … sides, since then pairs of faces can be parallel, and there can be an unambiguous “top” number. The cube can be seen as a special case of a trapezohedron.

For 12, 16, 20, …. sides, bipyramids make good dice (the octahedron is a special case of a bipyramid):

A bipyramid makes a good d16 – photo by “Traitor”

These are not the only shapes satisfying the definition, however. The 13 Catalan solids also satisfy it, and some of them make good candidates for dice. For example, the deltoidal icositetrahedron and the tetrakis hexahedron are both good candidates for d24:

The deltoidal icositetrahedron and tetrakis hexahedron are alternatives for d24 – photos by Jacqueline de Swart (left) and “Traitor” (right)

Some Catalan solids, like the pentagonal icositetrahedron, are unsuitable as dice because there is no unambiguous “top” number. On the other hand, there are some additional variations that are isohedral, like the hexakis tetrahedron.

For more on this subject, see Alea Kybos’ impressive dice page.

Geometry 1900 years ago

Papyrus Oxyrhynchus 29 (not to be confused with New Testament Papyrus 29) is a papyrus from the Oxyrhynchus collection, containing the statement of Proposition 5 of Book 2 of Euclid’s Elements, with an accompanying diagram. In modern notation, the proposition is ab + (ab)2/4 = (a+b)2/4. Euclid states the proposition as follows (the first paragraph is on the papyrus):

If a straight line be cut into equal and unequal segments, then the rectangle contained by the unequal segments of the whole together with the square on the straight line between the points of section is equal to the square on the half.

PROOF: For let a straight line AB be cut into equal segments at C and into unequal segments at D; I say that the rectangle contained by AD, DB together with the square on CD is equal to the square on CB.

For let the square CEFB be described on CB, and let BE be joined; through D let DG be drawn parallel to either CE or BF, through H again let KM be drawn parallel to either AB or EF, and again through A let AK be drawn parallel to either CL or BM.

Then, since the complement CH is equal to the complement HF, let DM be added to each; therefore the whole CM is equal to the whole DF.

But CM is equal to AL, since AC is also equal to CB; therefore AL is also equal to DF. Let CH be added to each; therefore the whole AH is equal to the gnomon NOP.

But AH is the rectangle AD, DB, for DH is equal to DB, therefore the gnomon NOP is also equal to the rectangle AD, DB.

Let LG, which is equal to the square on CD, be added to each; therefore the gnomon NOP and LG are equal to the rectangle contained by AD, DB and the square on CD.

But the gnomon NOP and LG are the whole square CEFB, which is described on CB; therefore the rectangle contained by AD, DB together with the square on CD is equal to the square on CB. Therefore etc. Q. E. D.

The papyrus is in Greek capitals; in modern letters it reads like this:

Modern scholars date the fragment to AD 75–125. It is not of great quality, with poor handwriting, spelling errors (μετοξὺ for μεταξὺ, and τετραγώνου for τετραγώνῳ on the last line), and missing labels on the diagram (making it of limited use, and perhaps explaining why it was found in an ancient trash pile). However, unlike the New Testament with its hundreds of manuscripts, there is not much of Euclid before AD 900, which makes this fragment historically very significant. It contains one of the oldest extant Greek mathematical diagrams.

See more here.