Classification of four-dimensional simplices with regular Morley simplex

Determine whether, up to similarity, the regular 4-simplex and the two nonregular classes $U_1$ and $U_2$ are the only 4-simplices whose Morley simplex is regular.

Background

The paper constructs two nonregular four-dimensional examples, U1U_1 and U2U_2, in addition to the regular simplex. Their existence and algebraic descriptions are established exactly.

These constructions do not provide a complete classification. The paper explicitly leaves open whether any further similarity classes of 4-simplices with regular Morley simplex exist.

References

These examples do not settle the classification in dimension four.

— Morley Simplices in Higher Dimensions: Regularity, Reflections, and Counterexamples  (2610.01216 - Tran, 1 Oct 2026) in Section 5.1, first item of Section 8, “Open problems”

For fixed $n\ge3$, are there only finitely many similarity classes of $n$-simplices with regular Morley simplex?

— Morley Simplices in Higher Dimensions: Regularity, Reflections, and Counterexamples  (2610.01216 - Tran, 1 Oct 2026) in Section 8, item 5, “Open problems”

Which of these simplices can be expressed by radicals?

— Morley Simplices in Higher Dimensions: Regularity, Reflections, and Counterexamples  (2610.01216 - Tran, 1 Oct 2026) in Section 8, item 7, “Open problems”