Euler's characteristic formula for convex polyhedra relates vertices ($V$), edges ($E$), and faces ($F$) by $V - E + F = 2$. Which of the following regular polyhedra satisfy Euler's formula with the correct tuple $(V, E, F)$?
Options:
(A) Regular Tetrahedron: $(V=4, E=6, F=4)$
(B) Regular Hexahedron (Cube): $(V=8, E=12, F=6)$
(C) Regular Octahedron: $(V=6, E=12, F=8)$
(D) Regular Dodecahedron: $(V=12, E=30, F=20)$
Answer Key:
A, B, C
Step-by-Step Solution & Explanation
**Method 1: Polyhedral Combinatorics**
- (A) Regular Tetrahedron: $V = 4, E = 6, F = 4 \implies 4 - 6 + 4 = 2$. True.
- (B) Regular Hexahedron (Cube): $V = 8, E = 12, F = 6 \implies 8 - 12 + 6 = 2$. True.
- (C) Regular Octahedron: $V = 6, E = 12, F = 8 \implies 6 - 12 + 8 = 2$. True.
- (D) Regular Dodecahedron has $12$ pentagonal faces ($F = 12$) and $20$ vertices ($V = 20$). The given tuple swaps $V$ and $F$ ($V=12, F=20$ describes an icosahedron). False.
Therefore, options **A**, **B**, and **C** are correct.