Triangle Dissection--Similar Triangles

A triangle dissection into similar triangles has mutually similar pieces, which need not be congruent or similar to the original triangle.

For a fixed triangle , let denote the number of dissections of into triangles, each similar to . Two dissections are identified when an isometry mapping to itself takes the collection of triangular regions of one to that of the other. The order of the tiles and the order of their vertices are ignored. Tiles may be directly similar or inversely similar, and their scale factors need not be equal. This differs from a rep-tile dissection, in which the smaller copies are required to be congruent.

A triangle is called generic here when no additional relation among its angles or sides makes otherwise distinct arrangements coincide or produces exceptional arrangements. The computed counts for several generic and special classes are summarized below (E. Pegg, Jr., pers. comm., Sep. 28, 2026).

class	OEIS	counts beginning at
generic non-right triangle	A399765	1, 0, 0, 4, 0, 36, 64, 468, 1792, 9262
generic isosceles triangle	A400510	1, 0, 0, 2, 0, 11, 19, 108, 385, 1747
generic right triangle	A399770	1, 1, 2, 10, 60, 363, 2346, 16581
Kepler triangle		1, 1, 2, 10, 60, 363, 2345
isosceles right triangle	A400511	1, 1, 1, 5, 20, 100, 543, 3209, 19986
30-60-90 triangle	A399999	1, 1, 3, 18, 121, 844, 6397, 53145

The first value counts the uncut triangle. A 3-4-5 right triangle realizes the generic right triangle counts. A computational enumeration based on dissection topologies by B. Trease, together with an independent coordinate search, verified the displayed generic right values (Pegg 2026). The 4-5-6 triangle, 45-60-75 triangle, and heptagonal triangle have the generic non-right counts through ten pieces (E. Pegg, Jr., pers. comm., Sep. 28, 2026).

The 60 five-piece dissections are illustrated above with coloring by B. Trease (Pegg 2026). The altitude from the vertex of the right angle to the hypotenuse gives the unique two-piece dissection, and subdividing either resulting triangle in the same way produces the two three-piece layouts in the enumeration. More generally, Zak (2005) proved that every triangle can be dissected into triangles similar to it for or .

If the pieces need only be mutually similar, their shape can differ from that of the original triangle. Zak (2005) proved that exactly nine shapes of triangle, up to similarity, admit a dissection into five mutually similar non-right triangles. Among non-right triangles, only the isosceles triangle with angles 30, 30, and 120 degrees admits five pieces similar to the original.

A 4-5-6 triangle admits two five-piece dissections into mutually similar triangles with side lengths in the ratio (Pegg 2019). These pieces are not similar to the original, so the constructions do not contribute to .

The three exact nine-piece constructions illustrated above have a single interior vertex at the first Brocard point, second Brocard point, or symmedian point. For the outer vertices , , and , the corresponding barycentric coordinates are , , and . Exact area and intersection calculations verify that each construction covers the 3-4-5 right triangle by nine nonoverlapping triangles similar to it (Pegg 2026).

For a 30-60-90 triangle, Pegg et al. (to appear) found that the numbers of arrangements with , 2, ... pieces are 1, 1, 3, 18, 121, 844, 6397, 53145, ... (OEIS A399999). This sequence was a runner-up in the OEIS 400K celebration contest (OEIS Foundation Inc. 2026). The 18 four-piece arrangements use 11 distinct multisets of piece areas. The 121 five-piece arrangements use 47 distinct such multisets, while the 844 six-piece arrangements use 235.

An arrangement is independent if it cannot be obtained from another cataloged arrangement by subdivision or by a local rotation or reflection. Among the five-piece arrangements, 84 arise by subdivision and 31 more have such a local partner, leaving 6 independent arrangements. The corresponding six-piece counts are 667, 154, and 23. There are 104 independent seven-piece arrangements. The 53145 eight-piece arrangements use 8793 distinct multisets of piece areas and include 731 independent arrangements. Altogether, the catalog through eight pieces contains 60530 arrangements (E. Pegg, Jr., pers. comm., Sep. 27, 2026). A complete search cannot be restricted to edge-to-edge triangulations: some valid layouts contain T-junctions, while others have an interior fan at which several right angle vertices meet without any triangle side continuing through the junction (Pegg et al. , to appear).