Tilings by Triangles

The tiles of a tiling are called its Elements. The Order is the number of elements in the tiling.

A tiling is Perfect if its elements are all of different sizes. Otherwise the tiling is Imperfect.

Tilings of the Plane by Triangles

It is well known that any triangle shape, along with copies rotated 180 degrees can tile the plane.. All quadrilaterals can also tile the plane.

Coxeter [1973] has shown that the fundamental domain of the group of symmetries in the plane generated by three mirrors can be either an equilateral triangle (angles pi/3 pi/3 pi/3 or 60,60,60 degree), isosceles right triangle (angles pi/4 pi/4 pi/2 or 45 45 90 degree), or (angles pi/6 pi/3 pi/2 or 30,60,90 degree) half equilateral triangle. Except for rectangular and square fundamental regions, the fundamental domains of all symmetries of the plane are subsets of these triangles as provided by the root 2 and root 3 proportional systems.

Tilings of Squares by Non-Right Triangles

M. Laczkovich has shown that there are exactly three shapes of non-right triangles that tile the square with similar copies, corresponding to angles (pi/8,pi/4,5pi/8), (pi/4,pi/3,5pi/12), and (pi/12,pi/4,2pi/3) (Stein and Szabó 1994).

Tilings of Squares by Pythagorean Triangles (Right Triangles with integer sides)

In 'Making Squares from Pythagorean Triangles' Charles Jepsen and Roc Yang show a square exists that can be dissected into m Pythagorean triangles (right triangles with integer sides) if and only if m > 5. This is shown by displaying a square that can be dissected into five Pythagorean triangles, and observing that a dissection into m triangles implies a dissection of a (usually larger) square into m+ 1 squares. It then remains to be shown that no square can be dissected into 2, 3 or 4 Pythagorean triangles. The existence of such a dissection in the case m = 4 would imply the existence of a rational point (x,y) with x> 4 on a certain elliptic curve, and algebraic geometers have shown that the curve has no such rational points. In particular, given triangles of shape 1-2-sqrt(5) with no two the same size, tile the square. The best known solution has 8 triangles (Berlekamp 1999).[4]

Tilings by Isosceles Right Triangles and by Equilateral Triangles

The authors of "The Dissection of Rectangles into Squares" (Brooks, Smith, Stone & Tutte) extended the electrical tiling theory they developed for square dissections to the dissection of isosceles right triangles and equilateral triangles into similar triangles of different sizes.

Tilings by Isosceles Right Triangles

Isosceles right triangles of different sizes can be assembled together into larger isosceles right triangles. They can also tile other polygons such as squares and rectangles, and surfaces such as cylinders and tori.

A tiling of a polygon by isosceles right triangles is called Compound if it has a properly contained subdivided triangle or rectangle. Otherwise the tiling is Simple.

Arthur Stone dissected a square into 7 different isosceles right triangles. The size of each isosceles right triangle is represented by the length of a shorter side. This is conventionally an integer when the side is parallel to a side of the square. Jasper Skinner discovered later that isosceles right triangles can tile a square perfectly without dissecting a main diagonal. This and further discoveries led to a review of the theory by two of the original authors, C.A.B. Smith and W.T. Tutte and the publication of a joint paper {3} by all three.

Extended Bouwkampcode (XB code) is a concise notation, devised by G.H. Morley, describing a tiling by isosceles right triangles by listing the elements from top to bottom and from left to right. Each element is represented by an integer with the value 10n+d where n identifies the element's size and d its orientation. d is the direction (0-7) in which the element's right angle "points" expressed as the angle measured clockwise from the "up" direction divided by 45 degrees. The element's shorter side is n when d is odd and n√2 when d is even. The canonical orientation is determined in a similar manner to Bouwkampcode.

Option 1 (suggested by J.D. Skinner): The first element whose uppermost vertices lie on the same horizontal line segment may be prefixed by a minus sign.

Option 2: Each element may be followed by the (x,y) coordinates of the element's right-angled vertex.

The XB code of A.H. Stone's square depicted above is (option 1 only) -77 43 -30 24 23 -14 13, or (option 2 only) 77 0 7 43 7 3 30 3 3 24 5 1 23 7 1 14 6 0 13 7 0. In the latter case the triplets of numbers could be separated by commas.

Isomers (tilings with the same set of tiles) are ordered by their XB code elements (giving side length and orientation, not coordinates). The sets of isomers with the same order:side are ordered by the XB code elements of the first (or only) tiling in each set. For each order:side, tilings in the same set are allocated the first or next 2-letter id from a range associated with the catalogue.

Tilings by Equilateral Triangles

The authors of "The Dissection of Rectangles into Squares" proved that an equilateral triangle cannot be dissected into equilateral triangles all of different sizes (orientation ignored). At least two triangles will be of the same size. See David Radcliffe's post for a proof. However equilateral triangles can tile in an up direction or a down direction, if these are considered different as they are not congruent, even if at the same size, then a kind of 'perfect' tiling is possible.

The equilateral triangled triangle tiling on this page by W.T. Tutte is believed to be the lowest order (15 elements - 2 extra polar elements = 13/2 = 6 ½) of this kind, making it comparable to Duijvestijn's simple perfect squared square.

REFERENCES

1. 'The Dissection of Rectangles into Squares' by R.L. Brooks, C.A.B. Smith, A.H. Stone, and W.T. Tutte in Duke Mathematical Journal, vol. 7, pages 312-40, 1940.
2. 'The Dissection of Equilateral Triangles into Equilateral Triangles' by W.T.Tutte in Proceedings of the Cambridge Philosophical Society, Vol. 44, pages 464-82, 1948.
3. 'On the Dissection of Rectangles into Right-Angled Isosceles Triangles.' in J Comb. Theory Ser. B Vol 80, 2, November 2000, pp. 277-319(43). by Skinner J.D.; Smith C.A.B.; Tutte W.T.
4. Mathworld; Square Tiling
5. Laczkovich, M. "Tilings of Polygons with Similar Triangles." Combinatorica 10, 281-306, 1990.
6. Making Squares from Pythagorean Triangles Charles Jepsen, Roc Yang , The College Mathematics Journal, Vol. 29, No. 4 (Sep., 1998), pp. 284-288
7. Ivars Peterson's mathTrek - Squaring Circles