MTE3103 Geometry

Topic 3 Regular and Semi-regular solids

3.1 Synopsis

In topic 1 and 2, we have learned about the patterns of 2-dimension in a plane. In this topic
we will investigate some of the three-dimensional figures which can be constructed using
regular polygons. Platonic solids are regular solids with convex vertices, also known as
convex regular solid. Archimedean solids are semi-regular solids and they also have
convex vertices where as Kepler-Pointsot solids have concave vertices.

3.2 Learning outcomes

1. Reinforce and develop your knowledge of basic geometric concepts;

2. increase your skills and knowledge of techniques for accurate geometric constructions;
3. foster your appreciation of the role of geometry in history; and
4. relate the exploration of space within this unit and the primary and secondary
curriculum.

3.3 Conceptual framework

Regular and
Semi-regular
solids

Vertices, Faces Archimedean Kepler-Poinsot

5 Platonic solids
and Edges solids solids

3.4 Five Platonic solids

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Figure 3.4(1) below shows the five Platonic solids. These solids are called the
tetrahedron, cube (hexahedron), octahedron, dodecahedron and icosahedrons. These
names are derived from the Greek words for the number of faces for each of the solids.

3.4.1 Identify each of the Platonic solid

Could you identify and name each of the Platonic solid in Figure 3.4(1) below? Try
to identify them according to the type of polygon that make up the faces, the
number of faces of the solids (tetrahedron, cube, octahedron, dodecahedron and
icosahedrons)

Figure 3.4 (1)

Platonic solids are clasified in the polyhedral group. Polyhedrons are solids whose
faces are plane polygons. Polygons make up the faces of the platonic solids. The
faces meet at the edges. The points where three or more edges meet are called the
vertices.

Think
All the Platonic solids have faces made up of regular polygons. Only five regular
solids are possible. Why?

3.4.2 Relationship between Platonic solids and the elements in nature

Platonic solids were discovered and known during Plato period (427 – 347 B.C.).
However, Platonic solids were not discovered by Plato, but are named after him
because of the studies he and his followers made of them. Plato also believed that

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there is a mystical correspondence between four of the solids and the four
‘elements’ as follows:

Cube earth
Tetrahedron fire
Octahedron air
Icosahedron water

and that dodecahedron envelops the entire universe.

3.4.3 Definition of Polyhedral

All polyhedra are three – dimensional where their faces are made up of plane
polygons. In Greek ‘poly’ means many and ‘hedron’ many faces. A regular
polyhedron is a polyhedron whose faces are all the same regular polygon. In other
words, the faces are made up of from one type of polygon only – and this is similar
to regular tessellation in topic 1.

Activity 3.4(1) : Investigating how many polygons can make one

vertex

1. Cut out a regular polygon, let us try equilateral triangle as a template.

Adakah anda dapat membentuk pepejal?
2. Using this equilateral triangle as template, try to construct a regular polyhedron
Kemudian cuba dengan bentuk segiempat sama dan pentagon. Adakah
net. Try to construct a net made up of six equilateral triangles as shown below.
anda dapat membina pepejal 3-dimensi berbucu cembung, jika sudut adalah
Cut out the net and try folding it in different ways until you are satisfied that we
sama dengan 3600?
cannot construct a convex three–dimensional figure with six or more equilateral
triangles meeting at a vertex.

We cannot construct a convex three–dimensional figure with six or more equilateral

triangles meeting at a vertex because the angle of an equilateral triangle is 600, so
3. We 0cannot0 construct a convex three–dimensional figure with six or more
6 x 60 = 360 . But, to construct a concave 3-dimensional figure, we can use more
equilateral triangles meeting at a vertex. Why?
than six faces that is more than 360 0. So, in order to construct a convex three–
dimensional figure, the number of equilateral triangle meeting at one vertex is three,
four or five as shown below.

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Tetrahedron:
Three equilateral triangles at one
vertex: 3 x 600 = 1800

Octahedron:
Four equilateral triangles at one
vertex: 4 x 600 = 2400

Icosahedrons:
Five equilateral triangles at one
vertex: 5 x 600 = 3000

Cube:
Three squares at one vertex:
3 x 900 = 2700

Dodecahedron:
Three pentagrams at one vertex:
3 x 1080 = 3240

Observe that to construct a convex regular polyhedron; the total angles at one vertex
must be less than 3600. That is why; there are only five convex regular polyhedral or
five Platonic solids.

Proof:
(i) The total angle of faces meeting at one vertex is less than 3600.
(ii) At each vertex at three equilateral triangles meet at one vertex and this can
be represented by the Schlafli symbol (3,3). Schlafli symbol (p,q) means
that the polyhedron has faces which are regular p-sided polygons, with q
polygons meeting at each vertex.

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Schlafli symbol (p,q)

p-sided regular polygon number of polygons meet at one vertex

Activity 3.4(2): Constructing Platonic solids

You are to construct each type of Platonic solids. Use an appropriate net. You may
get the nets needed from internet.
After you have constructed all the Platonic solids, observe and make an analysis of
their faces, edges and vertices. Complete Table 3.4(1).

Draw all the possible nets for each type of Platonic solids.

Suggestion: To construct an attractive and interesting solid, print the net on a colorful
designed paper. You can use the tessellation design that you have created
in topic 1.

Surf the internet to look for materials related to Platonic solids.

Solids

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Name of solid

No. of faces
meet at one
vertex

Schlafli
symbol (p,q)

No. of faces
(F)

No. of
vertices (V)

No. of edges
(E)

Dual

Table 3.4 (1)

3.4.4 The duals of the Platonic solids

Refer to Table 3.4(1) and you should see that there is a close relationship between
the Schlafli symbols (p,q) of each Platonic solid. For example, Schlafli symbol for a
cube is (4, 3) and that of an octahedron is (3, 4). The numbers of edges for both
solids are the same that is 12. The number of faces for a cube is the same as the
number of vertices of an octahedron and vice versa, So, we can say that the dual of
a cube is an octahedron and vice versa. The same with dodecahedron and an
icosahedron. The Schlafli symbol (p,q) for dodecahedron is (5,3) dan icosahedrons
is (3, 5). The numbers of edges for both solids are 30. So dodecahedron is the
dual of an icosahedrons and vice versa. For tetrahedron we say it is self-dual.

3.5 Verices, Faces and Edges

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Now that you had produced the Platonic solids and you are to investigate the
vertices, faces and edges. Firstly, we know how many faces each solid (except
cube) has by its name. Nevertheless, take each solid and count the number of
faces. We can try to find an efficient way of counting the faces of polyhedron. For
example, if a dodecahedron is put on a flat table, we can see one face at the top,
one at the bottom, five ‘attached to’ the top and five ‘attached to’ the bottom, giving
1 + 1 + 5 + 5 = 12 altogether.

Counting the faces in this way will make you familiar with the solids and help you to
find the number of vertices and edges. Dodecahedron has 12 faces, each of which
is a regular pentagon, that is each face has 5 sides. So, if we counted each face
separately, we could get 5 X 12 = 60 edges altogether. But each edge on the
dodecahedron connects two faces, so counting all the edges means we are
counting twice. So there must be 60 edges ÷ 2 = 30 edges.

Each edge connects two vertices. So if we counted each edge separately we could
get 2 X 30 = 60 vertices. But, for the dodecahedron, three edges meet at each
vertex so we would have counted each vertex three times. So again, there must be
only 60 ÷ 3 = 20 vertices.

What are the general formula to count the number of faces, edges and
vertices of polyhedron?

3.6 Semi-regular solids

Archimedean solids are semi regular solids because these solids are formed by using two
or more regular convex polygons with equal edges as the faces, and the same
arrangement of polygons meeting at each vertex. The main characteristic of Archimedean
solids is that each face is a regular polygon and each vertex the polygons are repeated, for

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example in a truncated tetrahedron, the polygons meeting at one vertex is hexagon-

hexagon-triangle. The Archimedean solids are convex semi-regular solids.

There are 13 types Archimedean solids:

1. (3, 4, 3, 4) cuboctahedron

2. (3, 5, 3, 5) icosidodecahedron

3. (3, 6, 6) truncated tetrahedron (truncated means ‘slicing off)

4. (4, 6, 6) truncated octahedron

5. (3, 8, 8) truncated cube

6. (5, 6, 6) truncated icosahedron

7. (3, 10, 10) truncated dodecahedron

8. (3, 4, 4, 4) rhombicuboctahedron, (also called small rhombicuboctahedron)

9. (4, 6, 8) truncated cuboctahedron, (also called the great rhombicuboctahedron)

10. (3, 4, 5, 4) rhombicosidodecahedron,

11. (4, 6, 10) truncated icosidodecahedron,

12. (3, 3, 3, 3, 4) snub cube, snub cuboctahedron (snub means the process of
arranging a polygon with triangles)

13. (3, 3, 3, 3, 5) snub dodecahedron (snub icosidodcahedron).

Activity 3.6 (1): Constructing Archimedean solids

Get into goup of two and surf through the internet to get some Archimedean nets. You
are require to construct at least three Archimedean solids.
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Compare and contrast the five Platonic solids and Archimedean solids.
MTE3103 Geometry

The idea of Archimedean solids are actually produced by truncating the

vertices of Platonic Solids. Do extra reading on this. You may find the learning is
more fun on this topic.

3.7 Prisms and anti-prisms

3.7.1 Prisms

A prism consists of two copies of any chosen regular polygon (one becoming the
top face, and one becoming the bottom face), connected with squares along the
sides. By spacing the two polygons at the proper distance, the sides consist of
squares rather than just rectangles. At each vertex, two squares and one of the
polygons meet. For example, based on a 7-sided heptagon, is the heptagonal prism
(4, 4, 7).

Example of prism:

Other examples of prism:

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Triangular Prism (4,4,3) Pentagonal Prism (4,4,5) Hexagonal Prism (4,4,6)

3.7.2 Anti-prisms

An anti-prism also consists of two copies of any chosen regular polygon, but one is
given a slight twist relative to the other, and they are connected with a band of
alternately up and down pointing triangles. By spacing the two polygons at the
proper distance, all the triangles become equilateral. At each vertex, three triangles
and one of the chosen polygon meet.

Example: heptagonal anti-prism (3, 3, 3, 7).

Other examples anti-prisms

Pentagonal Antiprism Hexagonal Antiprism

Square Antiprism (3,3,3,4)
(3,3,3,5) (3,3,3,6)

3.8 Kepler-Poinsot solids

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 Kepler-Poinsot solids are regular non-convex polyhedron, with concave faces.

 Also known as regular star polyhedra

 All the faces are congruent (identical) regular polygons

 The number of faces meeting at each vertex are the same

 There are four type of Kepler-Poinsot solids.

(i) Small Stellated dodecahedron

 12 faces, 12 vertices, 30 edges

5
,5
 Schläfli symbol is { 2 }

 The dual is Great Dodecahedron

(ii) Great Stellated Dodecahedron

 12 faces, 20 vertices, 30 edges

 Schläfli is { , 3}

 The dual is the Great Icosahedron

(iii) Great Dodecahedron

 12 faces, 12 vertices, 30 edge

 Schläfli symbol is {5, }

 The dual is Small Satellated Dodecahedron

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(iv) The Great Icosahedron

 20 faces, 12 vertices, 30 edges

 Schläfli symbol is {3, }

 The dual is the Great Stellated Dodecahedron

Activity 3.7(1): Constructing Kepler-Poinsot solids

Get into goup of two and surf through the internet to get some Kepler-Pointsot nets.
You are require to construct two Kepler Poinsot solids. Decorate the solids.

-- Happy Studying and Good Luck!--

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