Triangle Dissection--Exceptional Shapes

Exceptional shapes for triangle dissections into pieces similar to the original triangle have arrangements that coalesce or additional arrangements beyond those of a generic triangle of the same class. A triangle is generic here when no additional relation among its angles or sides makes otherwise distinct arrangements coincide or produces exceptional arrangements. Counts identify layouts under isometries of the original triangle, allowing directly similar and inversely similar pieces with different scale factors.

Special relations among sides can change a count. For the Kepler triangle, two layouts that are distinct for a generic right triangle coalesce into a single configuration at the Kepler proportions, giving a seven-piece count of 2345 rather than 2346 (E. Pegg, Jr., pers. comm., Sep. 28, 2026).

The exceptional seven-piece dissection of the Kepler triangle is illustrated above. Five pieces meet at the interior point.

Pegg calls the following coalescence examples "convergent triangles" (E. Pegg, Jr., pers. comm., Oct. 8, 2026). Write for the short leg divided by the long leg, so the side lengths are in the ratio . Each equation in the table selects its unique real root in . Here is the number of pieces and is the number of pieces meeting at the coalesced interior point. The catalog gives exact constructions, without asserting minimality of or completeness of the listed shapes.

equation for
	7	5
	8	4
	8	5
	9	5
	9	5
	9	4
	9	5
	9	4
	9	5
	9	5
	9	5

The seven-piece case is the Kepler triangle, with side lengths , where is the golden ratio.

The two eight-piece cases illustrated above have side lengths and , respectively. The red interior points mark junctions at which four and five pieces meet, respectively.

Requiring points whose coordinates are rational functions of a shape parameter to coincide gives polynomial equations after denominators are cleared. The listed equations have integer coefficients, so their roots are algebraic numbers. A nonzero polynomial has only finitely many roots, explaining why each such condition selects isolated proportions.

Pegg uses the term "sporadic similar triangle" for a triangle that admits additional arrangements beyond those of a generic triangle of the same class (E. Pegg, Jr., pers. comm., Oct. 7, 2026). This terminology refers to a gain in arrangements relative to the generic count. For example, an extra three-piece arrangement occurs for a 30-60-90 triangle, whereas a generic right triangle has two three-piece arrangements. Coalescence of arrangements, as for the Kepler triangle, can instead reduce a count.

Pegg's October 2026 collection contains 1569 stored dissections of 106 special triangle shapes (E. Pegg, Jr., pers. comm., Oct. 22, 2026).

The following catalog lists representative dissections supplied by E. Pegg, Jr. (pers. comm., Oct. 7, 2026). Here is the number of pieces in the specified construction, and does not by itself assert minimality or a complete enumeration of exceptional shapes. Write for the golden ratio, for the supergolden ratio, for the plastic constant, and for the tribonacci constant. The remaining parameters are the unique positive real roots of and of .

side ratios
	3
	5
	6
	6
	6
	7
	7
	7
	8
	8
	8
	9
	9
	9
	11
	11

The nine-piece example is a numerical witness. The other rows have exact constructions, but minimality of the first exceptional count has not been established for the two eleven-piece examples. The parameter is distinct from the superplastic ratio , which satisfies .

For the examples with side lengths that are powers of a common base, Pegg's label -abc means side lengths , so the digits are exponents of squared side lengths. For example, -012 means and -014 means . Several bases are self-summing ratios.

An additional example has side lengths and admits a nine-piece dissection into similar triangles (E. Pegg, Jr., pers. comm.). Its piece areas, relative to the original triangle, are , , four copies of , and three copies of . These sum to 1. The construction does not establish a first exceptional count for this shape.

Pegg also reports a cube root family with side lengths , where and is an integer, with pieces (E. Pegg, Jr., pers. comm.). The communicated examples have , 3, and 4, with 7, 9, and 11 pieces, respectively. The law of cosines gives an included angle of 120 degrees between the sides of lengths 1 and . A general exact construction remains to be supplied. This family differs from the exact integer family with side lengths and pieces for positive integers . Neither formula asserts minimality. For example, in the cube root family gives with 19 pieces, while the catalog contains a seven-piece construction of the same shape.