Kepler Triangle

The Kepler triangle is a triangle with side lengths in the ratio , where is the golden ratio. It is therefore a reciprocal proportion triangle. Surprisingly, it is also a right triangle, which follows from the identity

(1)

The Kepler triangle with side lengths , 1, and has angles

			(2)
			(3)
			(4)

and area

(5)

The altitude to the hypotenuse divides a Kepler triangle into two similar triangles with areas and times that of the parent. Repeatedly splitting every currently largest triangle in this way, while leaving all smaller triangles unchanged, produces at most two distinct sizes at any one stage. This is the , , case of the bound for this family of multiscale substitution tiling rules (E. Pegg, Jr., pers. comm., Oct. 19, 2026).

Successive altitude splits also give a six-piece dissection with relative child areas (two pieces), (three pieces), and (one piece). Replacing every currently largest triangle by this fixed six-piece rule gives three distinct sizes after the first refinement and five after the second, with at most five at any stage (E. Pegg, Jr., pers. comm., Oct. 20, 2026).

The number of dissections of a generic right triangle into seven smaller similar triangles is 2346. At the Kepler proportions, two generic layouts coalesce into a single configuration, reducing the count to 2345. The exceptional seven-piece dissection is illustrated above (E. Pegg, Jr., pers. comm., Sep. 28, 2026).