Irreptile

An irreptile is a bounded set in the plane, equal to the closure of its nonempty interior, that can be dissected into finitely many smaller similar copies of itself, with disjoint interiors. The copies may differ in size, and rotations and reflections are allowed. Under this inclusive convention, every polygonal rep-tile is an irreptile, while a dissection with unequal-size copies is also permitted (Richter 2009, p. 110).

If the linear scale factors of the pieces relative to the original are , ..., , then

For a rep-tile dissection into congruent copies, every scale factor is . The golden bee instead has a two-piece dissection with scale factors and , where is the golden ratio (Durand et al. 2020).

The figure shows selected chair-shaped polygons and their dissections, listed in reading order in the following table. The scale factors include repetitions and describe these particular dissections. The piece counts are not assertions of minimum order, and the examples do not constitute an exhaustive classification. Write for the silver ratio.

shape	pieces	linear scale factors
golden bee	2	,
Uyematsu chair	3	, ,
Miyoshi chair	3	, ,
Walton chair	3	, ,
L-triomino	4	, , ,
L-tetromino	4	, , ,
P-pentomino	4	, , ,
silver chair I	4	, , ,
silver chair II	4	, , ,

Pegg (2024a) reports that the Uyematsu chair was found by Mineyuki Uyematsu in March 2020. The Miyoshi chair is attributed to Jun-ichi Miyoshi (Pegg 2024b). The two silver chairs were found in a computer search by Miki Imura (2026). The three polyomino rows illustrate equal-size rep-tile dissections. The other six displayed dissections use unequal sizes. The two silver chairs have the same multiset of scale factors, whose squared values sum to 1 by .