Exploring Truncated Platonic Solids and Polyhedra Patterns
truncated platonic solids
what happens if you cut off all of the vertices
symmetrically, a little bit of the way along each of
the edges ?
what happens to the numbers of
•
faces
•
vertices
•
edges ?
tetrahedron
F
= 4
V
= 4
E
= 6
cube
F
= 6
V
= 8
E
= 12
octahedron
F
= 8
V
= 6
E
= 12
dodecahedron
F
= 12
V
= 20
E
= 30
For each 3D shape,
can you find out the
number of edges
faces and vertices
when we cut of the
vertices?
Truncated
tetrahedron
F
= ?
V
= ?
E
= ?
Truncated
octahedron
F
= ?
V
= ?
E
= ?
Truncated
dodecahedron
F
= ?
V
= ?
E
= ?
Truncated cube
F
= ?
V
= ?
E
= ?
icosahedron
F
= 20
V
=
12
E
= 30
truncated icosahedron (football)
F
=
?
V
=
?
E
=
?
what relationships are there between the ‘platonic’
values and the ‘truncated’ values?
Can you find the link between
(a) E and
F’
(b) E and
V’
(c)
E and
E’
Now fill in this table with all of your results
F
= 5
V
= 6
E
= 9
F’
=
F
= 5
V
=
5
E
= 8
F’
=
trying other polyhedra (does the pattern still work?)
V’
=
E
‘=
V’
=
E’
=
Discover what happens when vertices are symmetrically cut off from Platonic solid shapes, leading to changes in the numbers of faces, vertices, and edges. Explore relationships between the original Platonic values and the truncated values, and investigate similar patterns in other polyhedra shapes.
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Presentation Transcript
truncated platonic solids what happens if you cut off all of the vertices symmetrically, a little bit of the way along each of the edges ? what happens to the numbers of faces vertices edges ?
cube Truncated tetrahedron Truncated cube tetrahedron F = 6 V = 8 E = 12 F = ? V = ? E = ? F = 4 V = 4 E = 6 F = ? V = ? E = ? For each 3D shape, can you find out the number of edges faces and vertices when we cut of the vertices? dodecahedron octahedron F = 12 V = 20 E = 30 F = 8 V = 6 E = 12 Truncated dodecahedron Truncated octahedron F = ? V = ? E = ? F = ? V = ? E = ?
icosahedron F = 20 V = 12 E = 30 truncated icosahedron (football) F = ? V = ? E = ?
Now fill in this table with all of your results platonic tetrahedron F, V, E 4, 4, 6 truncated F , V , E cube 6, 8, 12 octahedron dodecahedron 8, 6, 12 12, 20, 30 icosahedron 20, 12, 30 what relationships are there between the platonic values and the truncated values? Can you find the link between (a) E and F (b) E and V (c) E and E
trying other polyhedra (does the pattern still work?) F = 5 V = 6 E = 9 F = V = E = F = 5 V = 5 E = 8 F = V = E =
