Squared Torus

A squared torus is a torus tiled by finitely many squares. The torus is regarded as a rectangle with opposite sides identified by translations, so a tile crossing one side continues at the opposite side. The squaring is called perfect when all the squares have different side lengths, and its order is the number of squares.

The opposite sides are identified without stretching the original rectangle, so its tiles retain their square geometry. Wrapping the diagram onto a doughnut-shaped ring torus, however, stretches some parts and compresses others. The resulting curved patches show which tiles meet, but do not preserve lengths or areas. Thus the tiles are squares in the planar geometry with opposite sides identified, even though their images on a ring torus do not look square.

This is a toroidal counterpart of a perfect square dissection. A Mrs. Perkins's quilt also dissects a square into smaller squares, but allows repeated side lengths and requires their greatest common divisor to be 1. For a squared torus, opposite sides of the fundamental domain are identified and tiles may cross its boundary.

Gambini (1999) constructed a perfect squared torus of order 24 with a square as its fundamental domain.

S. Anderson (pers. comm., Sep. 10, 2026) found the example illustrated above, of order 22 with a square as its fundamental domain. In these constructions the tile sides are parallel to the sides of the fundamental domain. Anderson's example has both smaller side length and lower order than Gambini's, but does not establish a minimum order among perfect squared tori with a square fundamental domain and tile sides parallel to its sides. Anderson's example consists of squares of sides 1, 3, 5, 6, 8, 10, 11, 14, 15, 18, 19, 22, 24, 27, 29, 30, 33, 34, 37, 38, 67, and 75. The squares of sides 24 and 67 cross the left-right and top-bottom boundaries, respectively, so the planar diagram has 24 pieces representing 22 squares on the torus.

Gardner (1966, 1977) asked readers how completely a square could be packed using the 24 squares having side lengths 1, 2, ..., 24. The number 70 is the side length of the containing square, not the order of the dissection. A perfect dissection, if one existed, would have order 24. Although

no perfect planar tiling exists (Bitner and Reingold 1975, Sgall et al. 2024). Gardner reported receiving about 250 submissions. The best left an area of 49 uncovered by omitting the square. In contrast, identifying opposite sides permits the following perfect squared torus.

S. Anderson (Pegg 2026) subsequently found a perfect squared torus of the same order 22 with a square as its fundamental domain. Its squares have side lengths 1, 2, ..., 20, 35, and 41, satisfying

The coordinates give an exact tiling with no gaps or overlaps after opposite sides are identified. This improves the side length of Anderson's example without reducing its order or establishing minimality.

The catalog compiled by G. Morley and reproduced by Anderson (n.d.) includes squared tori whose tiles are not parallel to the sides of the fundamental domain, including a perfect example of order 2 with a square as its fundamental domain. These belong to a broader class than the constructions with tile sides parallel to the sides of the square fundamental domain described above.