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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 andSemi-regular

solids

5 Platonic solids Vertices, Facesand Edges

Archimedeansolids

Kepler-Poinsotsolids


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3.4 Five Platonic solids

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 facesof the platonic solids. The

faces meet at the edges. The points where three or more edges meet are called the

vertices.

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

Think

All the Platonic solids have faces made up of regular polygons. Only five regular

solids are possible. Why?


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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.

Adakah anda dapat membentuk pepejal?

Kemudian cuba dengan bentuk segiempat sama dan pentagon. Adakah

anda dapat membina pepejal 3-dimensi berbucu cembung, jika sudut adalah

sama dengan 3600?

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

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

2. Using this equilateral triangle as template, try to construct a regular polyhedron

net. Try to construct a net made up of six equilateral triangles as shown below.

Cut out the net and try folding it in different ways until you are satisfied that we

cannot construct a convex threedimensional figure with six or more equilateral

triangles meeting at a vertex.

3. We cannot construct a convex threedimensional figure with six or more

equilateral triangles meeting at a vertex. Why?


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We cannot construct a convex threedimensional figure with six or more equilateral

triangles meeting at a vertex because the angle of an equilateral triangle is 60 0, so

6 x 600 =3600. But, to construct a concave3-dimensional figure, we can use more

than six faces that is more than 3600. 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.

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.

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


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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

polygonsmeeting at each vertex.

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.


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Solids

Name

of solid

Tetrahedro

n

Cube Octahedro

n

Dodecahedro

n

Icosahedrons

No. of

faces

meet atone

vertex

3 3 4 3 5

Schlafli

symbol

(p,q)

3,3 4,3 3,4 5,3 3,5

No. of

faces

(F)

4 6 8 12 20

No. of

vertice

s (V)

4 8 6 20 12

No. of

edges(E)

6 12 12 30 30

Dual Self-dual octahedro

n

cube Icosahedrons Dodecahedro

n

Table 3.4 (1)

3.4.4 The duals of the Platonic solids


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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

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 andvertices of polyhedron?


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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

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 (truncatedmeans 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 smallrhombicuboctahedron)

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

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)


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

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:

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.

Compare and contrast the five Platonic solids and Archimedean solids.

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.


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Other examples of prism:

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 ofalternately 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
http://home.comcast.net/~tpgettys/prism6.wrlhttp://home.comcast.net/~tpgettys/prism5.wrlhttp://home.comcast.net/~tpgettys/prism3.wrl


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Square Antiprism (3,3,3,4)Pentagonal Antiprism

(3,3,3,5)

Hexagonal Antiprism

(3,3,3,6)

3.8 Kepler-Poinsot solids

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 dod ecahedro n

12 faces, 12 vertices, 30 edges

Schlfli symbol is { 5,2

5}

The dual is Great Dodecahedron

(i i) Great Stellated Dodecahedr on


Schlfli is { , 3}

The dual is the Great Icosahedron
http://home.comcast.net/~tpgettys/anti6.wrlhttp://home.comcast.net/~tpgettys/anti5.wrlhttp://home.comcast.net/~tpgettys/anti4.wrl


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(i i i) Great Dodecahedr on

12 faces, 12 vertices, 30 edge

Schlfli symbol is {5, }

The dual is Small Satellated Dodecahedron

(iv) The Great Icosahedro n


Schlfli symbol is {3, }

The dual is the Great Stellated Dodecahedron

-- Happy Studying and Good Luck!--

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.

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