Problem
Dual polyhedron from face centers
Put one dot at the middle of every face of a solid. Join two dots whenever their faces share an edge. The dots and segments form a new solid inside. Say what comes out of a tetrahedron and a cube.
Reasoning · Grade 6-1 Dual and Semiregular Polyhedra
Your answer
Watch out The new solid looks like a shrunken copy, but faces and vertices swap.
How to solve
Strategy Draw a Diagram — Trying to see the inner solid through the outer one is hard, so instead I find the rule that turns the old counts into the new counts, and then read the answer off the numbers. Each dot sits on a face, so faces turn into vertices. Each segment sits across an edge, so the edge count does not change at all. And each face of the new solid wraps round one old vertex, so vertices turn into faces. Written as one line: faces and vertices swap places, edges stay put. I start with the tetrahedron, which is the easier case because you can see every dot at once, and use it to check the rule; then I apply the rule to the cube, where the inner solid is hidden.
1STEP 1
Where do the new vertices come from?
Each new vertex comes from an old face.
(new vertices) = (old faces)
This is just a matching: one dot for one face, with none left over on either side, which is exactly what makes two counts equal.
4.G.A.1Draw A Diagram2STEP 2
Where do the new edges come from?
Each new edge pairs with an old edge.
(new edges) = (old edges)
Each segment crosses one old edge like a little bridge over it, so pairing up segments and old edges leaves nothing over on either side.
4.G.A.1Draw A Diagram3STEP 3
Where do the new faces come from?
Each new face comes from an old vertex.
(new faces) = (old vertices), (sides of a new face) = (faces at an old vertex)
Cutting a corner off a solid leaves a small flat polygon, and the ring of dots round that corner is the same polygon, just drawn a bit further in.
4.G.A.1Visualize Spatial Relationships4STEP 4
State the rule
So faces and vertices trade places.
f ⇔ v, e → e
A rule that only swaps two numbers and leaves the third alone is easy to hold on to, and easy to check: Euler's v + f - e = 2 still comes out the same because swapping v and f does not change v + f.
6.EE.B.6Make A Systematic ListThe rule is that the new solid swaps the face count with the vertex count and keeps the edge count the same.
Why?
One dot is placed for one face and one new segment crosses one old edge, so each count is carried across one for one.
🧱One-to-one correspondenceIf each thing here pairs with exactly one thing there, the two groups are the same size.
Why?
Swapping the vertex and face counts leaves vertices minus edges plus faces untouched, so the new solid still obeys the rule every solid obeys.
🧱Euler's formula for solidsFor any solid with flat faces, vertices minus edges plus faces always comes out to 2, so a count you cannot see can be recovered.
5STEP 5
Part (1): the regular tetrahedron
A tetrahedron gives back a tetrahedron.
f = 4, e = 6, v = 4 ⟶ v' = 4, e' = 6, f' = 4
This is the case you can check by eye, because all four dots are visible at once. Getting the familiar answer back here is what makes the rule trustworthy for the cube, where you cannot see inside.
5.G.B.4Solve An Easier Related Problem6STEP 6
Part (2): the cube
A cube gives an octahedron.
f = 6, e = 12, v = 8 ⟶ v' = 6, e' = 12, f' = 8
The counts alone pin down the answer, because among the regular polyhedra only one has 8 faces, 12 edges and 6 vertices. And the top-bottom-four-sides picture of the six centres confirms the shape without needing the counts at all.
5.G.B.4Visualize Spatial RelationshipsAnswer
a tetrahedron, an octahedron
6 → 8, 8 → 6
Euler's formula holds for both new solids: the tetrahedron gives 4 + 4 - 6 = 2 and the octahedron gives 6 + 8 - 12 = 2, so the counts are those of a genuine polyhedron. The rule is also self-consistent — apply it twice and you should get back where you started, and you do: cube (6, 12, 8) becomes octahedron (8, 12, 6), and the octahedron becomes (6, 12, 8), the cube again. The tetrahedron is its own partner, which is exactly what you would expect from a solid whose face count and vertex count are already equal. Sizes are sensible too: the new solid always sits strictly inside the old one, since every dot is inside a face, and it never has more edges than the original, since edges are matched one to one.
Takeaway
Dots in the middles of the faces build a new solid where faces and corners trade places and the edges stay the same in number — so a cube turns into a regular octahedron, and a regular tetrahedron turns back into itself.
- Where do the new vertices come from?
- Where do the new edges come from?
- Where do the new faces come from?
- State the rule
- Part (1): the regular tetrahedron
- Part (2): the cube
You might solve next