Download 14. regular polyhedra and spheres

14. REGULAR POLYHEDRA AND SPHERES
14 - 1 Making polyhedra from regular triangles
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14 - 2 Making polyhedra from other regular polygons
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14 - 3 Regular and irregular polyhedra
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14 - 4 The Platonic solids
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14 - 5 Euler's formula
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14 - 7 Symmetry in Platonic solids
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14 - 8 Spheres
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14 - 9 Molecular geometry
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14 - 10 Living Platonic solids
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Activity 14 – 1
Making polyhedra from regular triangles
You made an equilateral triangle.
It is regular because all its sides are the same length and all its angles
are equal.
The part of the circle that is between a chord and the circumference of
that circle is a segment.
An octahedron has 8 faces.
“tetra” means 4
“octa” means 8
“icosa” means 20
Faces
Vertices
Tetrahedron
4
4
Faces at
each vertex
3
Octahedron
8
6
4
12
Icosahedron
20
12
5
30
Edges
6
If you try to make a solid in which 6 corners meet at every vertex, the
faces lie flat because 6 angles of 60°each make 360°. 360° is the sum of
angles at a point on a flat surface.
No. No more solids can be made using only regular triangles (ie.
equilateral triangles) because the sum of the angles at each vertex
would be greater than 360° and this is impossible.
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Activity 14 – 2
Making polyhedra from other regular polygons
The corners of a 4-sided regular polygon
Sum of the angles of a quadrilateral = 360°
If it is regular, the size of each angle = 360° ÷ 4 = 90°.
A solid in which 3 square faces meet at each vertex is called a cube.
A cube has:
6 faces, 8 vertices, 3 faces meeting at each vertex and 12 edges.
No. You could not make another regular polyhedron by putting corners
of 4 squares together at every vertex because 4 x 90° = 360° and this is
the sum of angles at a point on a flat surface.
The corners of a 5-sided regular polygon.
Sum of the angles of a pentagon = 180° x 3 = 540°
If it is regular, the size of each angle = 540° ÷ 5 = 108°.
"Dodeca" is a Greek word meaning 12.
A dodecahedron has:
12 faces, 20 vertices, 3 faces meeting at each vertex and 30 edges.
No. You could not make another regular polyhedron by putting corners
of 4 regular pentagons together at every vertex because 4 x 108° = 432°
and this is more than the sum of angles at a point on a flat surface.
No regular polyhedron can be made with faces that are regular
hexagons because:
Sum of the angles of a hexagon = 180° x 4 = 720°.
If it is regular, the size of each angle = 720° ÷ 6 = 120°.
The smallest number of faces that can meet at a vertex is 3.
3 x 120° = 360° and this is the sum of angles at a point on a flat surface.
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Activity 14 – 3
Regular and irregular polyhedra
The faces of a soccer ball are regular hexagons and pentagons.
4 faces
6 faces
8 faces
20 faces
The figure above that is crossed out is not a regular polyhedron because
at two of its vertices there are 3 faces that meet and at the other three
vertices, 4 faces meet.
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concave
concave
convex
If a polyhedron is concave, it cannot have all its dihedral angles (ie. the
angles between its faces) equal. To be concave, a polyhedron must
have at least one dihedral angle that is a reflex angle. For all dihehral
angles to be equal, they must all be reflex angles. If this happened, the
faces could not all meet each other.
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Activity 14 – 4
The Platonic solids
Fire (dryness) was represented by the Platonic solid with the smallest
volume for its surface area. The Platonic solid with the fewest vertices
has the smallest volume for its surface area. This solid is the regular
tetrahedron.
Water (wetness) was represented by the Platonic solid with the largest
volume for its surface area. The Platonic solid with the largest number of
vertices has the greatest volume for its surface area. This solid is the
regular icosahedron.
Earth was thought to be stand firmly (ie. perpendicularly) on its base.
This Platonic solid is the cube.
The universe was represented by the Platonic solid with 12 faces
because the zodiac has 12 signs. This solid is the regular
dodecahedron.
Air is mobile. It was represented by the Platonic solid that rotated most
freely when held by 2 opposite vertices. This solid is the regular
octahedron.
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Activity 14 – 5
Euler's formula
Each slice adds one extra face. Each slice also adds:
• 2 extra vertices
• 3 extra edges.
Polyhedron
Number of
faces (F)
Number of
vertices (V)
Number of
edges(E)
F+V
A
6
8
12
14
B
7
10
15
17
C
8
12
18
20
D
9
14
21
23
The number of faces plus vertices is 2 more than the number of edges.
F+V=E+2
Platonic
NUMBER OF:
Vertices Edges Vertices
Faces
(V)
(E)
on a face at a vertex
4
6
3
3
Tetrahedron
Faces
(F)
4
Octahedron
8
6
12
3
4
Icosahedron
20
12
30
3
5
Cube
6
8
12
4
3
Dodecahedron
12
20
30
5
3
solid
Yes. Euler's formula does apply to the Platonic solids.
The number of faces of the octahedron is the number of vertices of the
cube and vice versa.
The number of vertices on a face of the octahedron is the number of
faces at each vertex of the cube and vice versa.
The number of faces of the icosahedron is the number of vertices of the
dodecahedron and vice versa.
The number of vertices on a face of the icosahedron is the number of
faces at each vertex of the dodecahedron and vice versa.
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Activity 14 – 7
Symmetry in Platonic solids
In a regular tetrahedron:
3 planes of symmetry go through each vertex.
3 planes of symmetry go through each face.
One edges lies on each plane of symmetry.
It has 6 planes of symmetry altogether.
The axis of symmetry goes from a vertex, through to the centre of the
opposite face.
The regular tetrahedron turns through 120° before it looks identical to its
original position so its order of symmetry is 3.
The regular tetrahedron has 4 axes of symmetry.
All these axes of symmetry meet at the centre.
The planes of symmetry also meet at this point because each plane of
symmetry has an axis of symmetry that runs down its centre.
A regular octahedron has 7 planes of symmetry and 9 axes of symmetry.
All the planes and axes of symmetry meet at the centre.
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Activity 14 – 8
Spheres
A sphere has an infinite number of planes of symmetry.
A sphere has an infinite number of axes of symmetry.
Bubbles are spherical because this
shape requires the least amount of
surface area to cover a given volume
of gas.
A basketball is a sphere because this
gives it the same bounce no matter what
part of the surface hits the ground. It also
allows it to fit though a circular hoop no
matter what the orientation of the ball is.
Animals curl up into the shape of a
sphere to keep warm because this
minimises their surface area and
hence their exposure to cold air.
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Activity 14 – 9
Molecular geometry
This molecule has the shape of a
tetrahedron.
The isometric grid arrangement of balls takes up the least space.
The central atom is surrounded by 6 nearest neighbour
atoms (all equidistant from it).
These neighbouring atoms form the vertices of a
regular octahedron.
A central atom is surrounded by 8 nearest neighbour
atoms (all equidistant from it).
These neighbouring atoms form the vertices of a cube.
8 balls form the vertices of a cube.
The other 6 balls form a regular octahedron.
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Activity 14 – 10
Living Platonic solids
Viruses
This is an adenovirus that can
cause respiratory illness or
conjunctivitis.
It is a regular icosahedron.
Diatoms
A cube
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A regular dodecahedron
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