Exploring Geometry of Solids
Introduction
Solids are three-dimensional objects. In this chapter the discipline of geometry is used to explore and
describe the properties of solids of revolution and polyhedra.
The solids of revolution that will be explored in this and the next chapters are cylinders, cones, and
spheres.
The word polyhedron comes from the Greek, meaning an object with many faces. The faces of a
polyhedron are the polygons from which the polyhedron has been made. The edges of a polyhedron are
the sides of its faces. The vertices of a polyhedron are the vertices of its faces. Polyhedra are involved in
architecture, art, and nature. The shapes of many molecules and crystals are closely related to polyhedra,
such as the regular polyhedra, pyramids, and prisms.
The pyramids of Giza demonstrate an impressive level of geometry of polyhedra which was known to the
Egyptians of those times. However, our knowledge of their written mathematics is limited to two papyri,
Rhind Papyrus and Moscow Papyrus. Problem number fourteen of the Moscow Papyrus, written in
approximately 1890 BC, suggests a formula to compute the volume of a truncated square pyramid. Later
on, in Greece, Archimedes documented that Democritus, the Greek philosopher before him who lived
during the end of the fifth century BC, knew how to compute the volume of the pyramid.
The best-known polyhedra that have connected numerous disciplines such as astronomy, philosophy, and
art through the centuries are the regular polyhedra. They are known as the Platonic solids. There are five
regular polyhedra. By a regular polyhedron we mean a polyhedron with the properties that
(a) All its faces are congruent regular polygons.
(b) The arrangements of polygons about the vertices are all alike.
Platonic solids were known to humans much earlier than the time of Plato. There are carved stones (dated
approximately 2000 BC) that have been discovered in Scotland. Some of them are carved with lines
corresponding to the edges of regular polyhedra. Specifically, among them there is a dodecahedral form
that shows that the dodecahedron was known to the humans much earlier than it appears in any written
document. Icosahedral dice were used by the ancient Egyptians.
Evidence shows that Pythagoreans knew about the regular solids of cube, tetrahedron, and dodecahedron.
A later Greek mathematician, Theatetus (415 - 369 BC) has been credited for developing a general theory
of regular polyhedra and adding the octahedron and icosahedron to solids that were known earlier.
The name “Platonic solids” for regular polyhedra comes from the Greek philosopher Plato (427 - 347 BC)
who associated them with the “elements” and the cosmos in his book Timaeus. “Elements,” in ancient
beliefs, were the four objects that constructed the physical world; these elements are fire, air, earth, and
water. Plato suggested that the geometric forms of the smallest particles of these elements are regular
polyhedra. Fire is represented by the tetrahedron, earth the octahedron, water the icosahedron, and the
almost-spherical dodecahedron the universe.
The regular polyhedra are highly organized objects and have the most possible symmetry among all
polyhedra, which makes them aesthetically pleasing. These objects have enriched art and architecture
from the ancient times, to the Renaissance, to our modern times.
The semiregular solids are solids that have more than one type of regular polygon for their faces and in
each of them the arrangements of polygons about vertices are all alike. In addition to an infinite number
of regular prisms and anti-prisms, there are thirteen semiregular solids which are referred to as
Archimedean solids. Archimedes has been credited for the discovery and description of these thirteen
solids in a manuscript that is now lost.
Johannes Kepler proved the existence of the Archimedean solids and the two infinite families of prisms
and anti-prisms. He also discovered two non-convex regular polyhedra. Later, Lovis Poinsot (1777 1859) completed the work by finding the other two non-convex regular polyhedra. They are called four
regular stellated (star) polyhedra.
There have been and are mathematicians who have discovered more properties of polyhedra. To mention
a few, Rene Descartes (1596 - 1650) discovered a theorem about convex polyhedra that is equivalent to
the famous formula by Leonard Euler (1707 - 1783) that will be presented in this chapter. Norman
Johnson, in about 1960, conjectured that there are 92 regular-faced convex polyhedra in addition to the
regular and semiregular polyhedra. Later on, Johnson and other mathematicians, Grünbaum and
Zalgaller, proved this conjecture. There were more discoveries in recent years. However, these are
beyond the scope of this book.
7.1. Solids, Poly Pro, and Zome Geometry
A polyhedron is a finite region of space that is bounded by polygons. The polygons are called faces. The
faces intersect at edges. The points where three or more edges intersect are called vertices.
A convex polyhedron is a polyhedron that encloses a convex region; each segment created by two points
inside of this region stays entirely inside the region. From now on, by polyhedra we mean convex
polyhedra unless stated otherwise.
An important fact about the polyhedron is the existence of a relationship among its components that was
first discovered by Leonard Euler (1707 - 1783).
Vertices + Faces = Edges + 2
Euler was born in Switzerland and studied there until he accepted a position at the Russian Academy in
medicine and physiology. He then moved to the Berlin Academy in Prussia as a mathematician and
finally went back to the Russian Academy for the rest of his life. Euler is considered the greatest
mathematician of the eighteenth century, one who contributed to every area of mathematics known in his
time.
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In this section, we would like to explore some popular polyhedra.
The prism is a polyhedron with a pair of identical parallel faces called bases whose all other faces are
parallelograms. The altitude of a prism is any perpendicular segment that joins a point on a base to a
point on the plane containing the other base. If the bases are regular polygons, then the axis is defined as a
line that passes through the centers of the bases. A right prism is a prism whose bases are perpendicular
to the other faces. In case the bases are regular polygons, a right prism can be defined as a prism whose
axis contains an altitude. In this book we consider all prisms to be right prisms unless stated otherwise.
Axis
Axis
Altitude
Altitude
Figure 7.1.1. A prism and a right prism.
We define regular prisms as prisms with regular polygonal bases in such a way that all the surrounding
faces to the bases are squares. A regular prism is the cube (one of the Platonic solids).
Figure 7.1.2. A triangular prism, a regular square prism (cube), a square prism (cuboid), and a
pentagonal prism.
An anti-prism is a polyhedron with a pair of identical parallel bases where the surrounding faces are all
triangles. These triangles point up and down alternately. The axis and the altitude of anti-prisms are
defined the same as for prisms. Similar to the prisms, we consider all anti-prisms to be right anti-prisms
unless stated otherwise.
The regular anti-prism is an anti-prism with regular polygonal bases with surrounding congruent
equilateral triangles. A special case of the regular anti-prisms is the regular triangular anti-prism,
octahedron (one of the Platonic solids), which we will study in the next section.
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Figure 7.1.3. A regular triangular anti-prism (octahedron), a square anti-prism, a pentagonal antiprism, and a dodecagonal anti-prism.
A pyramid is a polyhedron that consists of a face, called the base, and surrounding triangular faces that
have one common vertex called the apex. The altitude is the perpendicular segment that joins the apex to
a point on the plane containing the base. In case the base is a regular polygon, the axis is defined as a line
that passes through the apex and the center of the base. A right pyramid is a pyramid whose altitude
joins the apex to the center of the base. In this book, we consider all pyramids to be right pyramids unless
stated otherwise.
Axis
Axis
Altitude
Altitude
Figure 7.1.4. A pyramid and a right pyramid.
A regular pyramid is a pyramid with a regular base whose all other faces are congruent equilateral
triangles. We notice, then, we have only three regular pyramids: the regular triangular pyramid
(tetrahedron—one of the Platonic solids), square pyramid, and the pentagonal pyramid (why not regular
hexagonal pyramid?).
Figure 7.1.5. A regular triangular pyramid (tetrahedron), a regular square pyramid, and a
regular pentagonal pyramid.
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Consider the patterns in Figure 7.1.6. We can cut out each of these shapes and then fold and fasten
together appropriately to create a cube (try it!). We can also study these patterns to explore some of the
properties that a cube possesses. Each of these patterns is called an unfolded cube or a net for a cube.
Figure 7.1.6
The German artist Albrecht Dürer (1471-1528) invented the concept of the net. To construct a net for a
polyhedron, we can use the Geometer’s Sketchpad. However, in more complicated cases than the cube,
we may need to have a three-dimensional model as an aid.
Here we would like to introduce two utilities that can improve our understanding of polyhedra by helping
us make their nets and models: Poly Pro software and Zome Geometry.
Poly Pro
This software creates an environment that allows us to observe, move, fold, and unfold certain groups of
polyhedra. It also generates the net of each of them. It includes an adjustable screen to see and move the
solid and a keyboard. The keyboard includes
the mouse.
and
for observing the solid and moving it using
The key
can generate the Schlegel diagram. A Schlegel diagram for a polyhedron is a twodimensional model for polyhedra. It presents what is viewed in perspective from a position outside the
center of one face, where this face appears as a frame and all the remaining edges are seen in its interior
(Figure 7.1.7).
Figure 7.1.7. The Schlegel diagram for the cube, the dodecahedron, and the icosahedron.
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The icon
creates the net of the polyhedra. The following are the nets for the regular triangular prism,
the regular square anti-prism, and the regular pentagonal pyramid generated in Poly Pro.
Figure 7.1.8
We also can use a key on the keyboard,
7.1.9).
, to fold and unfold a solid dynamically (Figure
Figure 7.1.9
Zome System
Zome system is a construction tool used to create geometric forms. It was primarily invented for scientists
and engineers for building abstract models for research. Using some Zome tools, we are able to discover
more relationships in solids than by just looking at pictures. Zome tools and a companion book, Zome
Geometry, distributed by Key Curriculum Press, provide many hands-on activities and instructions.
Zome system consists of two types of objects: struts and balls. Zome struts come in different colors and
sizes. Each color indicates a specific polygon for the cross section of a strut. The blue struts, B, have
rectangular bases. The yellow struts, Y, have triangular bases. The red, R, green, G, and green-blue, GB,
struts have pentagonal bases. The Zome ball is designed with appropriate holes in specific locations for
all these different struts.
Zome struts come in different sizes. We use subscripts to indicate the size of a strut. For example, B1
indicates the small blue strut, B2 is the middle one, and B3 is the large blue strut. It is not difficult to
discover that we can make a B3 size edge by connecting a B1 and a B2 in a straight line which shows that
B1 + B2 = B3. Similar equations are true for the yellow and red struts: Y1 + Y2 = Y3 and R1 + R2 = R3.
It also can be shown that B2/ B1 = B3/ B2 and therefore B2/ B1 = (B1+ B2)/ B2 (similarly Y2/ Y1 = (Y1+ Y2)/ Y2
and R2/ R1 = (R1+ R2)/ R2). This demonstrates that the struts for these colors illustrate the Golden
Proportion!
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Figure 7.1.10
There are other relationships among these struts, but we prefer to first construct some polyhedra and then
explore some of these properties through these constructions.
As a practice for prisms, let us construct a cube. This requires three-dimensional corners, each out of a
ball and three equal size struts. For a cube, three struts should be connected to a common ball in such a
way that each becomes perpendicular to the plane continuing the other two. We will soon discover that
the only possible case is using three equal size blue struts. Constructing a single three-dimensional corner
will show us how we complete our constructions of a cube.
Figure 7.1.11
Next, let us construct a triangular based prism. For instance, three B2s make an equilateral triangle for a
base. We also can make a triangle using two R2s and one B2. For the case of three B2s we need yellow
struts for the other edges of the prism (Figure 7.1.12.a). For the second case, we should use blue struts as
the surrounding edges (Figure 7.1.12.b). To build a right triangular prism, we need to use two B2s and
one G1 for each base of the prism and use the blue struts for the other edges (Figure 7.1.12.c).
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Y1
B1
B1
G1
R2
B2
B2
B2
R2
(a)
B2
B2
(b)
B2
(c)
Figure 7.1.12
To make a triangular pyramid, we have already discovered that three equal size Bs or two Rs and one B
make its triangular base. To complete the structure, we need to use three Rs for the first case, and two Bs
and one R for the second case. Then we see that these two structures are identical! Please note that an R
is a little shorter than its corresponding B. Therefore, the constructed pyramid (Figure 7.1.13.a) is not a
tetrahedron (the regular triangular pyramid). However, it is a good approximation for this Platonic solid.
To construct a tetrahedron we need to use six Gs. This is a difficult task due to the structure of the green
struts. To make this construction easier, you may first construct a cube using twelve B2s and then use six
appropriate green struts for its faces’ diagonals as is illustrated in Figure 7.1.13.b.
R2
R2
R2
G1
B2
B2
B2
B2
(a)
(b)
Figure 7.1.13
The solids of revolution are cylinders, cones, and spheres. Many everyday objects are shaped as the
solids of revolution.
A cylinder can be thought as a prism whose bases are two congruent polygons having infinitely many
sides. Its bases are congruent circles contained in parallel planes. The altitude is the perpendicular
segment that joins a point on a base to a point on the plane containing the other base. The axis is the line
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that joins the centers of the bases of a cylinder. A right cylinder is a cylinder whose axis is
perpendicular to the bases. Similar to the case for polyhedra, by a cylinder, we mean a right cylinder
unless otherwise stated.
A cone has a circular base and an apex that is not on the same plane as the base. The axis is the line that
joins the apex to the center of the base. The altitude is the perpendicular segment that joins the apex to a
point on the plane containing the base. A right cone is a cone whose axis is perpendicular to the base and
in this book we consider all cones as right cones unless otherwise stated.
A sphere is a set of points in space equidistance from a given point, called the center. A sphere is
generated by rotating a circle in space about one of its diameters. If a plane intersects a sphere in more
than one point, then the intersection will be a circle. If this plane passes through the center of sphere, the
circle is called the great circle. A great circle divides a sphere into two congruent hemispheres.
Exercise Set 7.1
1. What is a polyhedron? What is a convex polyhedron?
2. What is a prism? What is an anti prism? What is a regular prism and anti-prism?
3. What is a pyramid? What is a regular pyramid?
4. What is a cylinder? What is a right cylinder?
5. What is a cone? What is a right cone?
6. What is a sphere? What is a great circle on a sphere?
7. Figure 7.1.5 illustrates all the existing regular pyramids: Triangular, square, and pentagonal pyramids.
We notice that each pyramid is shorter in height compared to the preceding one. Write a short essay and
explain why we don’t have any other regular pyramid.
8. For each polyhedra illustrated in Figure 7.1.14 name the following:
I. All the faces.
II. All the edges.
III. All the vertices.
IV. Check the Euler formula.
D
D
G
F
L
E
K
H
F
A
B
(a)
Figure 7.1.14
K
E
E
F
D
A
C
H
G
C
A
C
B
B
(b)
(c)
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D
A
C
B
(d)
9. Complete the following table for polyhedra in Figure 7.1.14.
Polyhedron
Number of Faces
(F)
Number of Edges
(E)
Number of Vertices
(V)
Euler Formula
V+F=E+2
a
b
c
d
10. Complete the following table:
Polyhedron
F
E
V
Euler Formula
V+F=E+2
F
E
V
Euler Formula
V+F=E+2
F
E
V
Euler Formula
V+F=E+2
Pentagonal Pyramid
Hexagonal Pyramid
20-gonal Pyramid
n-gonal Pyramid
11. Complete the following table:
Polyhedron
Pentagonal Prism
Hexagonal Prism
20-gonal Prism
n-gonal Prism
12. Complete the following table:
Polyhedron
Pentagonal Anti-Prism
Hexagonal Anti-Prism
20-gonal Anti-Prism
n-gonal Anti-Prism
13. Use Zome tools to construct the following pyramids:
a. A triangular pyramid using three B2s and three R2s.
b. A square pyramid using four B2s and four Y2s.
c. A pentagonal pyramid using five B2s and five B1s.
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14. Use Zome tools to construct the following prisms and anti-prism:
a.
b.
c.
d.
A triangular prism using six B2s and three Y1s.
A square prism using eight B2s and four B1s.
A pentagonal prism using ten B2s and five R1s.
A pentagonal anti-prism using ten B2s and ten Y2s.
15. Use Poly Pro to construct
a.
b.
c.
d.
The octagonal prism,
The octagonal anti-prism,
The decagonal prism, and
The decagonal anti-prism.
Then, complete the following table:
Polyhedron
Number of Faces
(F)
Number of Edges
(E)
Number of Vertices
(V)
Euler Formula
V+F=E+2
a
b
c
d
16. Print a net for each polyhedron in Problem 15 and then construct its three-dimensional model.
17. Print a net for each regular pyramid and then construct its model. Please note that one of them is a
Platonic solid and the other two are Johnson solids.
7.2. Regular and Semiregular Polyhedra
Regular Polyhedra
We have already been introduced to the idea of regular polyhedra. In the introduction of this chapter we
characterized a regular polyhedron as a solid with the following properties:
1. All its faces are congruent regular polygons.
2. The arrangements of polygons about the vertices are all alike.
It was also mentioned that there are only five regular polyhedra and they are called the Platonic solids.
These solids were first explored by the Pythagoreans and were originally called the Pythagorean solids.
However, since Plato wrote Timaeus, a book that related the solids to the elements and cosmos, they are
now called Platonic solids. The Platonic solids and their associations with the elements and the cosmos
have been presented in Johannes Kepler’s book, Harmonices Mundi, as the following figures:
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Figure 7.2.1. The tetrahedron (fire), cube (earth), octahedron (air), icosahedron
(water), and dodecahedron (universe).
A question arises, “Why are there only five regular polyhedra?” To answer this, similar to the procedure
that was presented in Chapter Five for regular tessellations, we will investigate all possible cases of
arranging congruent regular polygons around a point. We start with equilateral triangles.
In a polyhedron, we define a vertex and surfaces around it as a three-dimensional “corner”. To create a
corner, we need at least three equilateral triangles. Moreover, from Chapter Five, we know that six
equilateral triangles fill the space around a point in a plane (they make a flat corner). Therefore, to create
a three-dimensional corner out of equilateral triangles, we need to put three, four, or five of them together
about a point.
We notice that the corner in Figure 7.2.2.a that is created by three equilateral triangles presents a solid
that is a regular polyhedron. This is because the generated base in this solid is another congruent
equilateral triangle. All the faces are congruent regular polygons, and the arrangements of polygons
about the vertices are all alike. More specifically, each vertex is surrounded by three equilateral triangles.
Similar to the symbol presentation of regular tessellations we can identify this regular solid as (p, q) = (3,
3). In this symbolism p presents the number of sides of each face, and q is the number of faces
surrounding a vertex. The (p, q) notation is called the Schläfli Symbolism, in honor of Schläfli, who
suggested it. This regular polygon is called tetrahedron (the four-faced regular polyhedron). Figure 7.2.3
presents a tetrahedron and its net, which was constructed in Poly Pro.
Now we take a look at Figure 7.2.2.b, which is a corner made from four triangles. The solid that presents
this corner is a square pyramid. Even though all the surrounding faces of this solid are congruent regular
polygons it is not a regular polyhedron due to the fact that the base is a different regular polygon (square).
However, if we glue two of these pyramids from their bases, then the created solid will be another
Platonic solid: octahedron (the eight-faced regular polyhedron). This regular polyhedron is constructed
from eight congruent equilateral triangles. Each vertex is surrounded by four triangles. Therefore, the
Schläfli symbol for this solid will be (3, 4). Figure 7.2.4 demonstrates an octahedron and its net using
Poly Pro.
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(a)
(b)
(c)
Figure 7.2.2. The constructions of three-dimensional corners out of (a) three, (b) four, and (c)
five equilateral triangles.
Figure 7.2.3. A tetrahedron and its net.
Figure 7.2.4. An octahedron and its net.
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Another possible case for a regular polyhedron with equilateral triangular faces is the case that each of its
vertices is surrounded by five triangles. Figure 7.2.2.c presents a corner of this solid. However, by just
looking at this corner is difficult to imagine the entire solid. Using Poly Pro we can see and study this
solid (Figure 7.2.5). It is the icosahedron (the twenty-faced regular polyhedron). Since each vertex is
surrounded by five triangles, the Schläfli symbol for this solid will be (3, 5).
Figure 7.2.5. An icosahedron and its net.
After equilateral triangles, we consider three-dimensional corners that are formed by squares. We need at
least three squares for a corner and from Chapter Five we know that four squares about a point make a flat
corner. Therefore, there is only one case for making a regular polyhedron out of squares: the cube. The
symbol for this solid is (4, 3). It is also called the hexahedron (the six-faced regular polyhedron). The
following figure presents the construction of a corner out of three squares, and also a cube and its net,
using Poly Pro.
Figure 7.2.6. A cube and its net.
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In Chapter Five we observed that three regular pentagons about a point leave a gap. This gap allows us to
construct a corner. Since four pentagons about a point overlap, the case of three pentagons for each
corner is the only case for creating a Platonic solid with pentagonal faces. Similar to the case for the
icosahedron, from a corner, it is almost impossible to imagine the entire pentagonal-faced regular
polyhedron. We use Poly Pro to construct this solid and present its net. It is called the dodecahedron
(the twelve-faced regular polyhedron). The Schläfli symbol for this solid is (5, 3).
Figure 7.2.7. A dodecahedron and its net.
When we place three regular hexagons about a point, as we noticed in Chapter Five, we create a vertex
point for the regular hexagonal tessellation, which is flat. Therefore, we cannot create a threedimensional corner out of three hexagons. For all other regular polygons with more than six sides, three
of each about a point will overlap (why?). Therefore, the only possible convex regular polyhedra are
three triangular-faced polyhedra of the tetrahedron, octahedron, icosahedron, one square-faced
polyhedron of the cube or hexahedron, and one pentagonal-faced polyhedron of the dodecahedron.
Similar to the definition of the dual of a regular tessellation in Chapter Five, we define the dual of a
regular polyhedron to be another regular polyhedron, which is formed by connecting the centers of the
faces of the original polyhedron (Figure 7.2.8).
The dual of the tetrahedron is the tetrahedron. Therefore, the tetrahedron is self-dual.
The dual of the octahedron is the cube.
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The dual of the cube is the octahedron.
The dual of the icosahedron is the dodecahedron.
The dual of the dodecahedron is the icosahedron.
Figure 7.2.8. The five Platonic solids and their duals.
The following table summarizes what we have studied about the regular polyhedra:
Table 7.2.1
Polyhedron
Tetrahedron
Hexahedron
Octahedron
Dodecahedron
Icosahedron
Schläfli Symbol The Dual
(3, 3)
(3, 3)
(4, 3)
(3,4)
(3,4)
(4, 3)
(5, 3)
(3, 5)
(3, 5)
(5, 3)
Number of Faces The Shape of Each Face
Equilateral Triangle
4
Square
6
Equilateral Triangle
8
Regular Pentagon
12
Equilateral Triangle
20
We may use Zome tools to construct each of these polyhedra:
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For the tetrahedron, as presented in the last section (Figure 7.1.12.b), we may use green struts. Also, a
good approximation for this solid can be obtained by using three blue and three red struts (Figure
7.1.12.a).
For the octahedron, we use twelve green struts. To form a vertex, four edges (struts) should meet at a
point (a ball). The green struts are not all the way straight. There are small bends on each end of a green
strut. We should connect four struts on a ball in such a way that the opposite struts have the same
orientations with respect to these bends. More specifically, two opposite struts with a shorter distance
between them on the ball should have a concave down bend near the ball and the other two struts should
have a concave up bend near the ball.
We are also able to approximate the octahedron with a triangular anti-prism. For this, we may use six
blue struts, B1, to construct two equilateral triangles for the bases and then use six red struts, R1, to
complete the structure.
To construct the icosahedron we need to use thirty blue struts. For this, first, construct two regular
pentagons. Then using these two pentagons construct a regular pentagonal anti-prism. The last step will
be to construct a regular pentagonal pyramid on top of each base.
For the cube, as we studied in the last section, we may use twelve equal sizes blue struts (Figure 7.1.10).
To construct the dodecahedron, we use thirty blue struts. We first form a regular pentagon. Then add one
strut on each ball in such a way that three struts which are connected to each ball make a three-fold
rotational symmetry with respect to the center of a triangular hole on the ball. The next step is to put a
ball on each of these struts and then add more struts on each ball to create the same three-fold rotational
symmetry on each ball. We continue this procedure to complete the shape.
Semiregular Polyhedra
Similar to semiregular tessellations, we define the semiregular polyhedra as follows:
A semiregular polyhedron is a polyhedron that holds the following properties:
1. Faces are of two or more types of regular polygons.
2. The arrangements of polygons about the vertices are all alike.
Unlike the tessellations, there are infinite number of semiregular polyhedra. However, if we group them
based on their shapes and properties, we will end up with only four groups.
The first group is the group of regular prisms. Each regular n-gonal prism is constructed from two regular
n-gons and n squares. Each vertex is surrounded by two squares and one regular n-gon. Therefore, each
regular prism is a semiregular polyhedron (Figure 7.2.9).
The second group of semiregular polyhedra is the regular anti-prisms. Each regular n-gonal anti-prism is
constructed from two regular n-gons and 2n equilateral triangles. Each vertex is surrounded by three
equilateral triangles and one regular n-gon. Therefore, each regular anti-prism is a semiregular
polyhedron (Figure 7.2.10).
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Figure 7.2.9. Some semiregular polyhedra from the fist group: A regular triangular prism, a
regular pentagonal prism, a regular hexagonal prism, and a regular octagonal prism.
Figure 7.2.10. Some semiregular polyhedra from the second group: A regular square anti-prism,
a regular pentagonal anti-prism, a regular hexagonal anti-prism, and a regular octagonal antiprism.
The third group has thirteen elements. They are called Archimedean solids. As was mentioned in the
introduction of this chapter, Archimedes has been credited for the discovery and description of these
thirteen semiregular polyhedra in a manuscript that is now lost.
Some of the Archimedean solids are obtained by truncating (cutting off) the corners of the regular
polyhedra. A few more are obtained by truncating these solids and then slightly distorting them.
If we truncate the corners of a tetrahedron by one-third of its edges, we will obtain our first Archimedean
solid: the truncated tetrahedron (Figure 7.2.11).
Figure 7.2.11. A truncated tetrahedron and its net.
Truncating each corner of a cube by one-third of its edges will result in another Archimedean solid: the
truncated cube (Figure 7.2.12.a). If with a similar procedure we truncate the corners by one-half of its
edges we will obtain the semiregular solid of cuboctahedron (Figure 7.2.12.c).
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(a)
(b)
(c)
(d)
Figure 7.2.12. (a-b) A truncated cube and its net, (c-d) A cuboctahedron and its net.
Another semiregular solid is obtained by truncating the corners of an octahedron by one-third of its edges.
It is called the truncated octahedron (Figure 7.2.13).
Figure 7.2.13. A truncated octahedron and its net.
If we truncate the corners of an octahedron by one-half of its edges, we will obtain the same solid as for
the procedure for the cube: the cuboctahedron. This interesting result is due to this fact that the cube and
the octahedron are each other’s duals. The following figure shows by properly truncating a cube, we will
obtain, first, the truncated cube, second, the cuboctahedron, third, the truncated octahedron, and finally,
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the octahedron (Figure 7.2.14). We also may start with an octahedron and properly cut the corners to
obtain the other three solids.
Figure 7.2.14
We are able to obtain two more semiregular solids by truncating a cuboctahedron and then slightly
distorting the shape:
If we truncate the corners of a cuboctahedron by one-third of its edges, we will obtain a shape whose
faces consist of regular hexagons, regular octagons, and rectangles. We imagine that with a slight
distortion these rectangles transform to squares. The result is the truncated cuboctahedron or great
rhombicuboctahedron (Figure 7.2.15.a).
For the second semiregular polyhedron, the rhombicuboctahedron, we truncate each corner of a
cuboctahedron by one-half of its edges and then distort the shape slightly by transforming the generated
rectangles to squares (Figure 7.2.15.c).
(a)
(b)
(c)
(d)
Figure 7.2.15. (a-b) A great rhombicuboctahedron and its net, (c-d) A rhombicuboctahedron and
its net.
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Similar to the cube and its dual, the octahedron, by truncating the corners of an icosahedron and its dual
dodecahedron, we will obtain truncated icosahedron (one-third truncation), the icosidodecahedron
(one-half truncation), and the truncated dodecahedron (one-third truncation).
Figure 7.2.16. An icosahedron, a truncated icosahedron, an icosidodecahedron, a
truncateddodecahedron, and a dodecahedron.
The following figure presents the nets for these three semiregular solids:
(a)
(b)
(c)
Figure 7.2.16. The nets for (a) the truncated icosahedron,(b) the icosidodecahedron, and (c) the
truncated dodecahedron.
With the same procedure as in the case for the cuboctahedron, we will obtain two more semiregular
polyhedra by truncation and then distortion of an icosidodecahedron as follows:
The one-third truncations of the corners and then transformation of rectangles to squares will result in the
truncated icosidodecahedron or great rhombicosidodecahedron (Figure 7.2.17.a).
The one-half truncations of the corners followed by the transformation of rectangles to squares will result
in the rhombicosidodecahedron (Figure 7.2.17.c).
(a)
(b)
(c)
(d)
Figure 7.2.17. (a-b) A great rhombicosidodecahedron and its net, (c-d) A rhombicosidodecahedron and its net.
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The two remaining Archimedean solids are the snub cube and the snub dodecahedron. The following
figure presents these two polyhedra and their nets.
(a)
(b)
(c)
(d)
Figure 7.2.18. (a-b) A snub cube and its net, (c-d) A snub dodecahedron and its net.
Similar to the semiregular tessellations we would like to introduce a notation for what regular polygons
meet at each vertex. For example, for a truncated tetrahedron we have a triangle and two hexagons about
each vertex. Therefore, this solid is identified by (3.6.6). The following table presents the Archimedean
solids’ notations:
Table 7.2.2
Archimedean Solids
Truncated Tetrahedron
Truncated Cube
Cuboctahedron
Truncated Octahedron
Great Rhombicuboctahedron
Rhombicuboctahedron
Truncated Icosahedron
Icosidodecahedron
Truncated Dodecahedron
Great Rhombicosidodecahedron
Rhombicosidodecahedron
Snub Cube
Snub Dodecahedron
Notation
3.6.6
3.8.8
3.4.3.4
4.6.6
4.6.8
3.4.4.4
5.6.6
3.5.3.5
3.10.10
4.6.10
3.4.5.4
3.3.3.3.4
3.3.3.3.5
The Archimedean solids can be inscribed in a regular tetrahedron so that four appropriate faces share the
faces of a tetrahedron. This means we always can find four faces in an Archimedean solid so that if we
extend them, then the intersections of these planes will form a tetrahedron.
The last group has only one element. This semiregular polyhedron does not have the above property of
the Archimedean solids. However, it can be obtained through a process from one of them:
If we cut off the top part of a rhombicuboctahedron and give it a one-eighth turn and then connect the two
pieces together, we will create the so-called semiregular polyhedron pseudo-rhombicuboctahedron.
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(a)
(b)
(c)
(d)
Figure 7.2.19. (a) A rhombicuboctahedron, (b) cutting off the top part,(c) one-eight turning of
the top part, (d) pseudo-rhombicuboctahedron.
Exercise Set 7.2
1. What is a regular polyhedron? How many regular polyhedra exist? Name them.
2. Why the regular polyhedra are called the Platonic solids?
3. How many triangular-faced regular polyhedra can be constructed? Explain why.
4. How many square-faced regular polyhedra can be constructed? Explain why.
5. How many pentagonal-faced regular polyhedra can be constructed? Explain why.
6. Why can’t we construct regular polyhedra with n-gonal faces, where n is 6 or a larger number?
7. What are semiregular polyhedra? Divide them into four groups and explain each group.
8. What are Archimedean solids?
9. Use Poly Pro to construct the regular polyhedra and then complete the following table
Polyhedron
F
E
V
Euler Formula
V+F=E+2
Tetrahedron
Hexahedron
Octahedron
Dodecahedron
Icosahedron
10. Use Poly Pro to obtain the net for each regular polyhedra and then construct them.
11. Use Poly Pro to obtain the net for each semiregular polyhedra and then construct them.
12. Using Zome tools construct the following regular polyhedra:
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e.
f.
g.
h.
i.
j.
k.
An approximation for tetrahedron using three B2s and three R2s.
A tetrahedron using six G2s.
A cube using twelve B2s.
An approximation for octahedron using six B2s and six R2s.
An octahedron using twelve G2s.
A dodecahedron using thirty B2s.
An icosahedron using thirty B2s.
13. Complete the following table:
Polyhedron Schläfli Symbol The Dual Number of Faces The Shape of Each Face
Tetrahedron
Hexahedron
Octahedron
Dodecahedron
Icosahedron
14. Using Zome system construct the following Archimedean solids:
l.
m.
n.
o.
p.
q.
r.
A truncated tetrahedron using 18 green struts and 12 balls.
A truncated cube using 12 blue struts and 24 green struts and 24 balls.
A cuboctahedron using 24green struts and 12 balls.
A truncated octahedron using 36 green struts and 24 balls.
A truncated icosahedron using 90 blue struts and 60 balls.
An icosidodecahedron using 60 blue struts and 30 balls.
A truncated dodecahedron using 90 blue struts and 60 balls.
15. Using Zome system construct a rhombicuboctahedron and a pseudo- rhombicuboctahedron. For this,
you need to use 24 blue struts, 24 green-blue struts and 24 balls.
16. Use Poly Pro to complete the following table:
Archimedean Solids
Truncated Tetrahedron
Truncated Cube
Cuboctahedron
Truncated Octahedron
Great Rhombicuboctahedron
Rhombicuboctahedron
Truncated Icosahedron
Icosidodecahedron
Truncated Dodecahedron
Great Rhombicosidodecahedron
Rhombicosidodecahedron
Snub Cube
Snub Dodecahedron
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Notation
3.6.6
3.8.8
3.4.3.4
4.6.6
4.6.8
3.4.4.4
5.6.6
3.5.3.5
3.10.10
4.6.10
3.4.5.4
3.3.3.3.4
3.3.3.3.5