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".
Shapes
Describe the geometrical shapes in the central square of the Vichten mosaic (depicted above).
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
Sizes
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.
Ratios
Describe the length ratio between the sizes L and M.
Describe the length ratio between the sizes XL and L.
Describe the length ratio between the sizes L and S.
Describe the remaining length ratios and the ratios for the areas.
The ratio between the lengths L and M is
=
.
To see this, compare octagons of size L and M by looking at the triangle between them.
The ratio between the lengths XL and L
= 1 +
can be deduced from the equality .
For this equality, notice that the L-sized octagon and two M-sized octagons cover the XL-sized square side.
The ratio between the lengths L and S is
= 2
can be obtained by combining the equalities and .
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.
The remaning length ratios can be computed as products of the ratios above or their inverse, for example .
The ratio for the areas is the square of the ratio for the lengths, for example
= 2.
