# Mathematical exploration of the Vichten Roman Mosaic The mosaic is on display at the Musée National d'Histoire et d’Art Luxembourg (MNHA).

Look here for a picture.

## What do you see, mathematically?

You should abstract from the artistic details and the thick frames. Just consider the mathematical plane shapes that form the central part of the mosaic.

• Describe the mathematical shapes in the central part of the mosaic.

Regular octagons, half square triangles, squares, pentagons (that are squares with a broken corner), eight-pointed star,...
Comment: The question "Which polygons do you see?" allows for different answers. Some may choose to list only the smallest figures (the disjoint pieces).

• Describe the symmetry of the central part of the mosaic.

Rotational symmetry (90 degrees), two perpendicular symmetry axes, central symmetry.

• Draw what you see in the central part of the mosaic. Make a qualitative sketch. Comment: Drawing an exact picture by hand requires some mathematical considerations and possibly some computations.

• Do you see a heart like this in the central part of the mosaic?  Comment: There are four (edgy) hearts at the four corners.

## Shapes and size

Look here for the mathematical representation of the central part of the mosaic.
• Compare the regular octagon to the octagon that it's easy to draw on a 3x3 grid. Draw them inscribed into a square (they are a square with broken corners).  In the former octagon you don't have a regular grid. In the latter octagon you break larger square corners.

• Assign sizes to the figures you see (Small/Medium/Large/Extra-Large). Assign the same size to an octagon, the square in which it is inscribed, and the remaining triangle corners.
• Extra-Large: the square frame and the inscribed octagon and the four corners.
• Large: the two central squares, the central octagon, the eight central triangles, four triangles at the square frame, and four triangles between them.
• Medium: the eight octagons on the sides, eight triangles at the square frame.
• Small: the squares at the corners of the square frame.
• Compare the ratios between figures of different sizes.
• The ratio between Large and Medium is √2. To see this, compare octagons looking at a triangle between them.
• The ratio between Extra-Large and Large is 1+√2. To see this, notice that the central octagon and two lateral octagons cover the square frame from left to right.
• The ratio between Medium and Small is √2. To see this, notice that the triangle corner of the square frame has the same height as the lateral octagon. This height is the sum of two legs of medium triangles and one leg of a large triangle. The height of the small square is one leg of a medium triangle and one leg of a large triangle.
• The ratio between Large and Small is 2. To see this, consider that the square frame is covered from left to right in two ways: two small squares and two medium squares; two small squares and one large square.

### Further explorations/Oncoming

• Stars: There are two different regular eight-pointed stars, and one can compare them. It is possible to draw one from the other with a geometrical construction that can be explored. One can have a glimpse into the mathematical classification of regular stars.
• Rhombus: Conside the famous rhombus with 45 degrees angles. Draw it with compass and ruler. Relate it to other figures (like the eight-pointed star).
• Pentagon: Classify interesting pentagons obtained by cutting one corner out of a square. Describe those present in the Vichten mosaic.
• Detail: Beyond the central part of the mosaics, there is an interesting pattern consisting of "whirls". The basic shape is a semidisc without two semidiscs. Between whirls one has some kind of diamond and some kind of heart, that can easily be investigated because their boundary consists of circular arcs. By counting tiles in the mosaic it seems that the hole in the middle of the whirl is 1/8 of the thickness of the central "knot".
• Tilings: Introduce the notions of periodic and aperiodic tilings. Show classical tilings and recently discovered tilings.
• Mosaic stones: Count the amount of mosaic stones in different parts of the mosaic to get information on lengths and areas. Construct (approximate) triangles that are half of a square in at least two ways out of small square pieces of the same size.
• Geogebra: Exercises with Geogebra. The mosaic drawn with Geogebra, where it is possible to highlight some figures. Examples of highlighted figures as a picture gallery.
• Polypad: The mosaic drawn with Mathigon's Polypad (https://mathigon.org/polypad). Invent other "mosaics" (tilings, possibly only with the shapes found in Roman mosaics).
• Other material: For a group work, produce mosaic pieces, for example cut them out of cardboard. Then reproduce a colorful version of Vichten's mosaic.