Tetrakis pentachoron

The tetrakis pentachoron, also known as the triangular-pyramidal icosachoron, is a convex isochoric polychoron with 20 triangular pyramids as cells. It can be obtained as the dual of the truncated pentachoron.

It can also be obtained as the convex hull of two dually oriented pentachora, where one has edges exactly $$\frac73 ≈ 2.33333$$ times the length of those of the other. Any convex hull of two dual pentachora where one is more than $$\frac32 = 1.5$$ times the edge length of the other gives a fully symmetric variant of this polychoron.

The ratio between the longest and shortest edges is 1:$$\frac{7\sqrt{19}}{19}$$ ≈ 1:1.60591. Each cell is a triangular pyramid with long base and short side edges.