The Cube, Built and Checked
Advanced companion to Cubo Stacks It Up, trading the character frame for the real vocabulary: a cube has 6 square faces, 8 vertices, and 12 edges, satisfying Euler's formula V minus E plus F equals 2. At each vertex three 90-degree angles meet for 270 degrees total, a 90-degree angular defect. Swapping the cube's face count and vertex count produces an octahedron (8 faces, 6 vertices): the cube and octahedron are dual solids.
Start with one square face, flat and steady on the ground, add a second square beside it, now two faces found, one, two, three squares standing, wrapping around the side, four squares closing the ring, no open gap left to hide, five squares now, the shape rises, one more square left to lay, six flat squares complete the cube, sealed the tidy way. Eight sharp corners hold the squares together at every seam, count the vertices, two, four, six, eight, closing out the scheme, twelve straight edges run between them, framing every face, six faces minus twelve edges plus eight vertices checks the space, eight take away twelve, plus six, lands exactly back at two, Euler's rule confirmed again, the same as every solid true. At each corner, three square angles meet, ninety degrees times three, two hundred seventy total, ninety short of a full sweep, room to be. Swap the cube's faces for corners and an octahedron comes into sight, six faces trade for eight, eight corners trade for six, a dual pair built right, two solids mirror one another, faces and corners neatly swapped, the cube and the octahedron, geometry's own matched.
Key Concepts: A cube has exactly 6 square faces, 8 vertices, and 12 edges, satisfying Euler's formula V − E + F = 2 | At each vertex, three 90-degree angles meet for 270 degrees, a 90-degree angular defect below a full 360-degree turn | Swapping the cube's face count and vertex count (6 and 8) produces an octahedron (8 faces, 6 vertices) — the cube and octahedron are dual solids
Characters: none — advanced companion sung about the real solid directly, paired with Cubo Stacks It Up
Key Words: cube, vertex, edge, face, Euler's formula, angular defect, dual solid, octahedron
Advanced framing note: The original Cubo Stacks It Up builds the six faces one at a time for a younger audience. This companion keeps the same building structure and adds the vertex/edge count, Euler's formula, the angular-defect fact, and the real cube-octahedron duality the original explicitly defers as "beyond this age range" — all standard 9-13 geometry-extension content.