Symmetric Canonical Self-Dual Tetrahedra

This is the only one

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Self-Dual Tetrahedron (canonical)
Vertices:  4  (4[3])
Faces:4  (equilateral triangles)
Edges:6  (all same length)
Symmetry:  Full Tetrahedral  (Td)
Dihedral Angle:  acos(1/3)    ≈70.528779366 degrees
(values below based on midscribed radius = 1)
Edge Length:  2*sqrt(2)    ≈2.8284271247461900976
Circumscribed Radius:  sqrt(3)    ≈1.73205080756887729353
Midscribed Radius:  1
Inscribed Radius:  sqrt(3)/3    ≈0.57735026918962576451
Volume:8/3    ≈2.6666666666666666667