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by Gareth Boreland
It is well known to students of basic geometry that the join to form a dodecahedron - we will see later why there
sum of the interior angles in a polygon with V vertices is must be 12. (It is also interesting to note that the resulting
solid has 20 vertices, each with an angular defect of 36', (V - 2) x 180'. Regular polygons will tessellate the plane if
their interior angle divides 360'. Thus the regular polygons giving a total defect of 720' - this is also discussed later.)
shown in the table below will tessellate. Let q be the number
of polygons meeting at each vertex of the tessellation.
Regular polygon Interior angle q
equilateral triangle 600 6
square 900 4
regular hexagon 120 0 3
The dodecahedron
Other regular polyhedra can be formed by 'closing the gaps'
in a similar way with other regular polygons.
Let q be the number of the regular polygons meeting at each
vertex.
Six equilateral triangles meet at a vertex to tessellate the
plane; the table shows the polyhedra formed when fewer
than six equilateral triangles meet at each vertex and the
'gaps are closed':
Regular hexagons will tessellate the plane
q Structure formed {p,q}
6 tessellation of plane Bright students will see immediately that since, after 120,
5 icosahedron {3,5}
the only remaining factors of 360 are 180 and 360, it is clear
4 octahedron {3,41
that no other regular polygons will tessellate.
3 tetrahedron {3,31
For example, in a regular pentagon each interior angle is
1080.
Now 360 = 3 x 108 + 36. So in attempting to tessellate
regular pentagons, we find that three will fit together, but
leaving a 36' 'gap' or 'angular defect':
The icosahedron
We can 'close the gap' by lifting two pentagons out of the flat
plane. By allowing them to fit together in this way, and
The octahedron
continuing this '3-D tessellation' we find that 12 pentagons
8 Mathematics in School, November 2007 The MA web site www.m-a.org.uk Cutting the corners off (i.e. truncating) an icosahedron
yields the traditional 'football' shape (a truncated
icosahedron) as shown:
The tetrahedron
The symbol {p,q} (called the Schlifli symbol) is used for a
polyhedron in which q regular p-sided polygons meet at each The centres of the faces of a cube form the vertices of a
vertex. regular octahedron, and vice versa. These two solids thus
have the same symmetries.
Similarly for squares:
A similar duality exists between the icosahedron and the
dodecahedron. q Structure formed {p,q}
4 tessellation of plane
One of the most remarkable results concerning polyhedra
3 cube {4,31
is surely Euler's Theorem. This states that, for any
convex polyhedron with F faces, V vertices and E edges,
F + V = E + 2. It can be instructive for pupils to investigate
this relationship and try to 'discover' it for themselves. For
the more complicated solids, pupils can use models to help
them count. Even then it is tricky, merely by counting, to
find, for instance, E for an icosahedron; pupils can be
encouraged to come up with mathematical relationships that
20 x 3
will help them - e.g. in this instance E-=2
The cube
2
Prisms and pyramids can be investigated easily and, once
For completeness we tabulate the result noted already for
models or diagrams of specific cases have been used,
regular pentagons:
students can be encouraged to find E, F and V for prisms
and pyramids with n-gonal bases:
(Note q cannot be 4 or more, since 4 x 108 > 360; there will
be no 'gaps' to close.)
E F V
n-gonal based prism 3n n + 2 2n q Structure formed {p1q}
n-gonal based pyramid 2n n + 1 n + 1
3 dodecahedron {5,3}
and finally for regular hexagons:
q Structure formed {p,q}
3 tessellation of plane
Clearly no other regular convex polyhedra can be
constructed in this way - we require q to be at least 3, but for
n-gons with n>6, 3 x interior angle > 360'. It is instructive
to see if students can be prompted to give similar arguments.
So we have found the five regular convex polyhedra, which
have been studied by mathematicians since about 350 BC
and which are known, of course, as the Platonic solids.
Apentagonal prism (E = 15, F = 7 and V = 10)
Other authors have spoken of the benefits to learners from
working with 3-D models of geometric results to be studied
(Olkun and Sinoplu, 2006) and I would agree that perhaps
the best way for students to understand the results described
above is to see, or even to build, models of them. Indeed,
many geometric facts concerning polyhedra are most easily
demonstrated by models; in fact some are difficult to
visualize without them. As examples, consider the following
results, each of which can grasp the attention and interest of
students:
SA cube has order 3 rotation symmetry about a body
A pentagonal pyramid (E = 10, F = 6 and V = 6) diagonal.
Mathematics in School, November 2007 The MA web site www.m-a.org.uk 9 Many proofs of Euler's Theorem exist. Tony Gardiner
180,(nF -2F )=360V- 720
(2003) presents a proof using spherical geometry. See also
n
David Eppstein's web page Nineteen Proofs of Euler's (multiplying by 360 and rearranging).
Theorem (wwwl.ics.uci.edu/- eppstein/junkyard/euler/)
which gives proofs based mostly on topology and graph
Thus S=180oF (n-2)= (V-2)x360.
n theory. There are proofs, however, which are accessible to
school students - see, for instance, www.wikipedia.org/wiki/
Tony Gardiner (2003) uses spherical geometry, rather than a
Euler characteristic.
prior assumption ofE + 2 = F + V, to derive results which
can be shown to be related closely to this. He proceeds to run
The theorem can be used to prove some of the results which
a similar argument in reverse to prove Euler's Theorem.
were merely stated before. For example, it was noted that 12
regular pentagons form the faces of a dodecahedron. Euler's
(It can also be noted that the result can be generalized for all
Theorem can be used to prove that the polyhedron {5,3}
polyhedra as S = (V- X) x 3600 where X = V + F - E is the
must have 12 faces.
Euler characteristic of the polyhedron.)
In this case we must have E = 5_ and V = 5F (because two
pentagons meet along each polyhedron edge and three
References
pentagons meet at each polyhedron vertex).
Gardiner, T. 2003 'The Euler Descartes Theorem'. In Pritchard, C. (Ed.)
The Changing Shape of Geometry, Cambridge University Press, pp. Then, using F + V = E + 2,
333-338.
5F 5F
Olkun, S. and Sinoplu, B. 2006 'Internalizing Three-Dimensionality
we obtain F+ 3F= --+2
Through Making Toys', Mathematics in School, 35, 5.
3 2
and so F = 12.
I conclude with a result that may not be so well known. We
have mentioned that in any polygon the sum of the interior
Keywords: Polyhedra; Euler; Platonic solids.
angles is given by (V - 2) x 180'. An analogous result exists
for the sum of all the face angles in a polyhedron.
Author
Gareth Boreland, Sullivan Upper School, Holywood, Co. Down BT18 9EP.
Theorem e-mail: [email protected]
In any convex polyhedron the sum of all the face angles, S,
is given by S = (V - 2) x 360', where V is the number of
vertices.
This result can be rewritten as 3600x V- S = 720'. This
shows that it is completely equivalent to Descartes' law of
angular defect. The angular defect at a vertex is defined to be
From Little Acorns
360' minus the sum of the face angles at that vertex.
by Lesley Ravenscroft, Diana Cobden,
Descartes' law states that the sum of the angular defects at all
Colin Abell and Elaine Griffin
the vertices of a convex polyhedron is 720'. We saw this
earlier in the case of a regular dodecahedron. I hope the way
The purpose of From Little
in which the result is stated here makes it more accessible to
ettle
Acorns is to share with teachers
students.
in Primary, Middle and Secon-
dary schools some of our ideas ra 9 Proof
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about the uses of Spreadsheets to
enhance the teaching of Mathe-
Let such a polyhedron have F faces, E edges and V vertices.
matics.
Spreadsheets 9 to 13
We especially wished to support Let Fn be the number of n-gonal faces. winu~irr r~ru~iur
the inexperienced Maths / I.T.
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1 teacher. And those interested in
Thus F = Fn andE =- 1nFn The Mathematical
developing the use of I.T. to n 2 n Association
support pupils' learning.
(The 1 is required because two polygons meet at each
polyhe2dron edge.)
Members Price 07.59
S=-YF x1800(n-2)=180 F (n - 2). Non Members Price 010.99 ISBN: 0 906588 36 7
n n
Now, the sum of the face angles on each n-gonal face is
MATHEMATICAL ASSOCIATION
Order from: (n - 2) x 180'. Thus the sum of all the face angles is given by
The Mathematical Association
Euler's Theorem is then applied: 259 London Road Leicester LE2 3BE
Telephone 0116 2210014
E+2-F+VI
E + 2 = F + V Email [email protected]
or order Online at www.m-a.org.uk supporting mathematics in education
- (nFn)+2=i (F)+V
2 n n n
10 Mathematics in School, November 2007 The MA web site www.m-a.org.uk