Mathematical exploration of the Vichten Roman Mosaic

The mosaic is on display at the Musée National d'Histoire et d'Art Luxembourg (MNHA)

Roman Sangaku

The central part of the Vichten Mosaic
is the tessellation of a square, and a Sangaku.

What is a Sangaku?

Sangakus are a special kind of geometry exercises that are often presented as pictures without text.
The unspoken convention in Sangakus is the following: "what looks regular, is regular".

Describe the geometrical shapes in the central square of the Vichten mosaic.

By considering only the disjoint pieces of the tessellation, we can identify the following shapes:

regular octagons (in two different sizes)
half-square triangles (in two different sizes)
symmetric pentagons that are a square with a broken corner

We also would like to mention:

an eight-pointed star in the middle of the square
an octagon inscribed in the square frame
squares at the corners of the square frame

Assign sizes to the geometrical shapes of the tessellation. Sizes: S=small, M=medium, L=large, XL=extra large.

Extra-Large: The square frame and its inscribed octagon are of size XL. The four triangles at the corners of the square are also of size XL (because they relate the octagon and the square of size XL).
Large: The central octagon and the two central squares which circumscribe the octagon are of size L. The related eight triangles forming the eight-pointed star are of size L. The four triangles at the middle of the sides of the square frame are also of size L.
Medium: The eight octagons surrounding the octagon of size L, and the eight triangles at the square frame are of size M.
Small: The four squares located at the corners of the square frame are of size S.

Describe the ratio between the sizes
L and M.

The ratio for the lengths is $\frac{L}{M}$ = $\sqrt{2}$ .
To see this, compare octagons of size L and M by looking at the triangle between them.

Describe the ratio between the sizes
XL and L.

The ratio $\frac{XL}{L}$ = 1 + $\sqrt{2}$ can be deduced from the equality $XL=L+2M$ .
For this equality, notice that the L-sized octagon and two M-sized octagons cover the XL-sized square side.

Describe the ratio between the sizes
L and S.

The ratio $\frac{L}{S}$ = 2 can be obtained by combining the equalities $XL=2M+2S$ and $XL=2M+L$ .
To get the two equalities, decompose the side of the XL-side square. Firstly, with two M-sized squares together with two S-sized squares. Secondly, with two M-sized squares and one L-sized square.

Describe the remaining length ratios and the ratios for the areas.

The remaning length ratios can be computed as products of the ratios above or their inverse, for example $\frac{M}{S} = \frac{L}{S} \frac{M}{L} = \sqrt{2}$ .
The ratio for the areas is the square of the ratio for the lengths, for example $\frac{M²}{S²}$ = 2.

Broken corners

We investigate octagons and pentagons that are "a square with broken corners".

When you draw an octagon on a 3x3 grid, do you obtain a regular octagon?

Left, we have an octagon at the 3x3 square grid.
Right, we have a regular octagon.
They are not the same.

Describe the octagon at the 3x3 grid.

The octagon at the 3x3 grid has four symmetry axes.
It has a 90° rotational symmetry.
There are two different side lengths, whose ratio is $\sqrt{2}$ .
The height is three times the shorter side lengths.
The area is 7/9 of the area of the square frame.

Compare the pentagon at the 2x2 grid and the pentagon of the Vichten mosaic.

The two pentagons are irregular. They are convex and symmetric.
They both have three 90° angles and two 135° angles.
The pentagon at the 2x2 square grid has three side lengths that are proportional to $1$ , $2$ , $\sqrt{2}$ .
The pentagon from the Vichten mosaic has three side lengths that are proportional to $1$ , $2$ , $1 + \sqrt{2}$ (it may help to see the pentagon as a quarter of a regular octagon).

Eight-pointed stars

There are two regular eight-pointed stars, namely the 8/2 star and the 8/3 star, see the figures below.
In the Vichten mosaic, there is an 8/2 star.

Describe the construction of the two regular eight-pointed stars inside a given a regular octagon.

The two stars consist of selected diagonals of the regular octagon.
The 8/2 star is given by the eight smallest diagonals, among vertices with 2 sides in between.
The 8/3 star is given by the eight intermediate diagonals, among vertices with 3 sides in between.

Draw an 8/3 star starting from an 8/2 star and conversely.

Starting from an 8/2 star, prolonge the eight lines. You get eight additional intersection points which are the vertices of an 8/3 star.
Starting from an 8/3 star, the segments in the interior of the star form an 8/2 star.

Braids pattern

In the central part of the Vichten mosaic,
different parts are separated by braids. It is interesting to analyze such braids at the merging points.

Invent a strategy to count the number of ropes seen in the braids pattern.

All ropes are closed. To count them we can do the following:

Begin with one rope, say in the upper-left corner, and follow its path until reaching its starting point.
Color the whole rope to mark that we already consider it.
Select another rope and repeat the process until the entire pattern is colored.

It seems that there are 6 ropes, including the central octagonal braid of the mosaic.

Here this picture as PDF.

Whirls pattern

In the Vichten mosaic, there are two rectangular stripes that contain a pattern made with a "whirl" shape.

Describe the whirl shape.

It is the union of four congruent figures that differ by rotations of 90°.
Each figure consists of two parts:
The first part is a half-circle without two half-circles (of half the size).
The second part is a half-annulus that continues as a rectangle.

Describe the optical illusion concerning the two "identical" stripes of whirls from the Vichten mosaic.

The two rectangular stripes have the same area.
The different number of whirls contained in the two rectangles is noticeable by comparing the orientation of the first and the last whirls in a row.
The orientation of the whirls alternates. In the stripe above (respectively, below) there is an odd (respectively, even) number of whirls.

Remarks

Here an article with the mathematical description of the Vichten Roman Mosaic.
The geometrical picture of the mosaic can be reproduced
- with Geogebra
- with Polypad
- by cutting cardboard (or another material)
- by 3D printing them.
The artistic decorations in the Vichten mosaic and the tessellation with mosaic tiles provide more geometrical shapes and mathematical activities. Other Roman mosaics showcase different geometrical shapes, for example the rhombus with 45° angles.

Many thanks to Marko Peric, Dany Alves Marques, Eduardo Rodrigues Da Costa, Maurice Desquiotz, João Rocha Figueiredo, Max Thill, Alexandre Benoist, Daniil Murzykaev.