Introduction
In geometry, there are special groups of polyhedra, many sided three dimensional
figures, Platonic solids is one of the most interesting cases. Platonic solids are five special
polyhedra. The five solids are “perfect” or regular polyhedra; all faces of each of the
Platonic solids are congruent regular polygons. Additionally, the same number of faces
that meet at one vertex. The five Platonic solids are the tetrahedron, cube, octahedron,
dodecahedron and icosahedron.
Figure 1: The five Platonic Solids (Definition of Platonic Solids, 2010)
We will be looking at three big ideas of Platonic solids that will help teachers create
lessons based upon these five solids. The three big ideas are: 1. the history of platonic
solids, 2. a proof why there are only five platonic solids, and 3. duals and the technology
behind them. These three topics will help students to explore and discover new ways
math can be used as well as connecting math and social studies together. An
understanding of these three ideas will truly help teachers and students alike, to know
how important platonic solids can be, as well to make connections in forming arguments
and data based on some simple premises.
History of Platonic Solids
Plato and the Platonic Solids
If you want to look at the history behind the five Platonic solids, you must go back
to ancient Greece. Well actually, you must go back a little further as the solids themselves
had appeared in various formats before reaching ancient Greece, including the stones
illustrated in Figure 2 below so in reality we do not know who exactly discovered the
solids, but we do know who they were named after.
Figure 2: Ancient Platonic Solids (Grier and Adele, 2003)
Fast forward to Greece, as a group of mathematicians called the Pythagoreans had
called these solids the Pythagorean solids, but they had only defined three of the solids,
but then Plato started to study them as well and he helped to defined the last two. The
solids were then named after Plato due to all the work that he did with them
(Encyclopedia Britannica, 2013).
Plato had used these five solids in his book called the Timaeus. Plato, in his book
Timaueus, went on to describe how the five solids represented the five basic elements
of the world: fire, air, water, earth and æther, better known as the universe.
Figure 3: Plato’s Elements (Earthegy, 2010)
The five basic elements of the world were assigned to a specific Platonic solid as Plato
used the overall shapes of the Platonic solids to determine which of the basic elements
to assign to each of the five Platonic solids. The first was the tetrahedron, which was
assigned fire, due to its sharp points and edges. He assigned the element of earth to the
cube because it had four square regularity, basically meaning that it had squares on all
sides. The octahedron was assigned to the element of air; the icosahedron was assigned
water as they were both made out of triangles. The last solid, the dodecahedron, was
assigned a special element; as Plato wrote , “There still remained a fifth construction,
which the god used for embroidering the constellations on the whole heaven.” (Timeaus,
5,6) Since the dodecahedron has 12 sides, and Plato, and many other greeks, used 12
different constellations to tell about the universe, Plato assigned the dodecahedron the
element of æther, or the universe (Encyclopedia Britannica, 2013).
Renaissance artists and Platonic Solids
History about the Platonic Solids is scarce beyond Plato naming them and basing
the elements of the universe on them. We do know they play into many different
Renaissance artists paintings. Many artists, such as Leonardo da Vinci, Paolo Uccello,
as well as astronomers and mathematicians, Johannes Kepler and Luca Pacoli, used the
Platonic Solids in their works as they were interested in the math behind them and the
symmetry shown in the solids (Sala, 2000 1-6). For example, Kepler used the Platonic
Solids to show the distance between the planets Mercury, Venus, Earth, Mars, Jupiter
and Saturn. Kepler proposed that the distance between the planets could be explained
by the platonic solids,as illustrated in Figure 4. The solids today are still used in art and
architecture because of their amazing symmetry.
Figure 4: Kepler’s model of planetary distances (Hart, 1998)
Why are There Only Five Platonic Solids?
Why are there only five platonic solids? Well there is a proof of why, but first we
need to look at a table that will help us along the way.
Platonic Solids
Type and number of faces meeting at a
vertex
Tetrahedron
3 triangles
Cube
3 squares
Octahedron
4 triangles
Dodecahedron
3 pentagons
Icosahedron
5 triangles
Table 1: Platonic Solids and how many faces meet at a vertex
This table is extremely important to our proof. The table shows how many faces meet at
a vertex, as well as what shape those faces are. Each of those faces are regular polygons,
which means they are equiangular and equilateral. The faces for platonic solids are only
made up of triangles, squares and pentagons. Why is that? It comes down to the interior
angle measure at the vertices.
Figure 5: Interior Angles (VB Helper, 2010)
Illustrated in Figure 5, interior angles are the angles are on the inside of polygons, so
when we talk about the interior angles at vertices, it is the total sum of the interior angles
of the polygons that meet at the vertice.
Regular Polygon
Angle Measures (in
degrees)
Interior Angle measure
when three faces meet at
a vertex (in degrees)
Triangle
60
180
Square
90
270
Pentagon
108
324
Hexagon
120
360
Heptagon
128.5
385.7
Table 2: Regular Polygons and their angles
As Table 2 suggests, regular polygons are equiangular when it comes to their
interior angles. Table 2 also suggests that in a Platonic solid (or any three-dimensional
figure) has to have at least three faces meeting at a vertex. Table 2 describes that if three
pentagons, squares, or triangles meet at a vertex, the total angle measure is less than
360°. Why is 360° so special? It is the total angle measure of a circle. A circle is a flat
two-dimensional object, that cannot be used to form a vertex on polyhedra. Circles do not
have vertices to begin as vertices are properties of both two-dimensional and threedimensional figures. The conjecture for the proof is in order to form a Platonic solid out of
regular polygons, then at a vertex the total sum of the interior angles that meet at a vertex
must be less than 360°. (Math is Fun, 2014).
Table 2 helps to confirm the conjecture, as it contains the angle measures for
triangles, squares and pentagons, as well as when three triangles, three squares and
three pentagons meet at a vertex. The Table contains the interior angle measures for
three of the platonic solids: tetrahedron, cube and the dodecahedron. The interior angle
measures of the other two platonic solids, octahedron and icosahedron are not contained
within Table 2; but both of the interior angle measures can be figured out since both have
triangles for faces. In an octahedron, four regular triangles meet at a vertex, and the
interior angle measure of a triangle is 60°, so the total interior angle measure at a vertex
in an octahedron is 240°. The interior angle measure at a vertex of an icosahedron has
the same interior angle measures since they are regular triangles. The only difference
between an icosahedron and an octahedron is that five triangles meet at a vertex, instead
of four. So the total interior angle measure when five regular triangles meet at a vertex is
300° (Math is Fun, 2014).
Now that we see that all the platonic solids meet the criterion of all interior angles
at a vertex is less than 360°. There is one more thing to address. If you try and make a
Platonic solid with any other regular polygon besides triangle, square or pentagons, then
it will not work. Going back to Table 2, if you look at if three regular hexagons or three
regular heptagons meet, than the total angle measure will be greater than 360°. If we
were to try and make a Platonic solid with any regular polygon pass the pentagon, then
the shapes would not form a vertex because the interior angle measure will be too big.
Therefore, there can only be five Platonic solids because if any regular polygon that has
more sides than a pentagon would try to form a Platonic solid, it would not be able to form
a vertex because of the angle measure would be greater than 360°. Vice versa, you can
only form Platonic solids out of regular triangles, squares and regular pentagons because
if 3 (or 4 or 5 if you look at triangles) faces made up of regular triangles, squares or
pentagons,than the angle measure would be less than 360° and able to form a vertex
(Math is Fun, 2014).
Duals: A lesson about Platonic Solids
Duals is the next big idea and in order to look at it, I want to connect it to the
classroom. If you look at the Common Core Standards that most states use, you can
directly connect Platonic solids and their duals to different standards such as being able
to draw, construct, and describe geometrical figures and describe the relationships
between them. Through duals, we can look at the relationships between these particular
solids as well explore congruence in regular polygons. Let us look at an example to help
define what duals are.
Figure 6: Dual of a cube and an octahedron (VB Helper, 2010)
Figure 6 gives a better image of what a dual is. Duals occur when the vertices of
the one Platonic solid, in this case an octahedron, replaces the faces of another, which in
Figure 6 is a cube.The edges of the Platonic solids replace each other as well. The best
way to start this lesson is to explore the three main properties about the Platonic solids:
faces, edges and vertices.
Platonic Solid
Faces
Edges
Vertices
Tetrahedron
4
6
4
Cube
6
12
8
Octahedron
8
12
6
Dodecahedron
12
30
20
Icosahedron
20
30
12
Table 3: Properties of Platonic Solids
As seen in Table 3,the faces, edges and vertices of Platonic solids are listed and confirms
that some of the Platonic solids are pairs of each other called reciprocals, in which two
Platonic solids can replace each other as a dual. The three properties will help students
to recognize patterns and notice that some of the solids would fit together and are paired
up together, but how?
You can do many different activities to show how, I think one of the best
visualizations is to use polydrons (seen in Figure 7), which are manipulatives that connect
together, and use them to make a dual and let their imaginations roll on what a dual is.
Figure 7: Polydron set of Platonic Solids (Educators Outlet, 2014)
These polydrons are a good concrete example to use as you can interact with the shapes
directly and have them physically in front of you. This could lead into a discussion about
what a dual is and the properties of the shapes, this can have students to think critically
and lead into the technological approach to duals.
Duals can be built online using a special program called Geogebra. Here you can
visually see the vertices of an octahedron replace the faces of the squares and vice versa.
Figure 7: Geogebra sketch of a Dual (Ben Dotson, 2014)
It would be nice to just let students try and make a dual without any guidance as learning
technology is a good way to introduce Geogebra as a tool for students to view and make
their own technological math tools. The lesson really can connect all that you learn about
platonic solids and rationalizing it to discover what duals are. Teachers can use duals to
help students with spatial reasoning and understand how the solids are connected.
Conclusion
Platonic solids are a great way to explore how math is connected to different topics
such as history and art. From the discovery of the solids by Plato, to their use in the
Renaissance, the solids were known for a very long time and their discovery made
mathematicians rethink three-dimensional shapes. The proof of why there are only five
Platonic solids help to connect angles and solids, and the type of shape and angle.A study
of duals helps students see how the solids are connected.
References
Dotson, B. (2014, October 1). Properties of a Cube and an Octahedron. Retrieved
December 5, 2014, from https://www.geogebratube.org/student/m161835
Educators Outlet, Buy More For Less. (2014, January 1). Retrieved December 5,
2014,
from
http://www.educatorsoutlet.com/index.php?main_page=index&cPath=10_93
Earthegy. (2010, January 1). Retrieved December 5, 2014, from
http://blog.earthegy.com/?p=1761
Grier, N., & Adele, G. (2003, October 10). The Geometry of Creation. Retrieved
December 5, 2014, from
http://www.webpages.uidaho.edu/ngier/polyhedra/goc.htm
Hart, G. (1998, January 1). Johannes Kepler's Polyhedra. Retrieved December 1,
2014, from http://www.georgehart.com/virtual-polyhedra/kepler.html
Heilbron, J. (2013, January 1). Platonic solid (mathematics). Retrieved
December 5, 2014, from
http://www.britannica.com/EBchecked/topic/464212/Platonic-solid
Hind, R. (1973). The Timaeus of Plato. New York: Arno Press.
MAINdictionary.html. (2014, January 1). Retrieved
December 5,
2014, from
http://intermath.coe.uga.edu/tweb/gwin101/apley/Dictionarycomments/welcome/MAINdictionary.html
Platonic Solids Definition. (2010, January 1). Retrieved December 5, 2014, from
http://www.formyschoolstuff.com/school/math/glossary/p.htm
Platonic Solids - Why Five? (2014, January 1). Retrieved November 24, 2014, from
http://www.mathsisfun.com/geometry/platonic-solids-why-five.html
Sala, N. (2002, January 1). Art, Mathematics and Architecture for Humanistic
Renaissance: The Platonic Solids. Retrieved November 13, 2014, from
http://math.unipa.it/~grim/SiSala2.PDF
VB Helper: Tutorial: Drawing Platonic Solids. (2010, January 1). Retrieved December 5,
2014, from http://www.vb-helper.com/tutorial_platonic_solids.html