Liceo Scientifico Isaac Newton

Maths course

PolyhedraPolyhedra

Professor

Tiziana De Santis

face vertex

edge diagonal

A A convex polyhedronconvex polyhedron is the part of space bounded by n is the part of space bounded by n

polygons (with n ≥ 4) belonging to different planes, so polygons (with n ≥ 4) belonging to different planes, so

that each edge of the polyhedron is the intersection of that each edge of the polyhedron is the intersection of

two of themtwo of them

Euler’s relationEuler’s relation

All convex polyhedra satisfy this important relation All convex polyhedra satisfy this important relation

between the numbers of faces (between the numbers of faces (FF), of vertices (), of vertices (VV) and of ) and of

edges (edges (EE))

32 + 60 – 90 = 2

F + V – E = 2F + V – E = 2

32 faces

2012

90 edges

60 vertices

Regular polyhedraRegular polyhedra

A polyhedron is said to be regular if its faces are regular and A polyhedron is said to be regular if its faces are regular and congruent polygons, and its dihedral angles and solid angles are congruent polygons, and its dihedral angles and solid angles are also congruentalso congruentThese solids are also called platonicThese solids are also called platonic

TetrahedronTetrahedron

It has four faces, four vertices and six edgesIt has four faces, four vertices and six edges

Three Three equilateral trianglesequilateral triangles converge in each vertex converge in each vertex

( Euler: 4 + 4 – 6 = 2 )

Symmetries:Symmetries:

- 6 planes- 6 planes passing through the passing through the

barycentre containing one edgebarycentre containing one edge

- 3 lines- 3 lines passing through middle passing through middle

points of opposite edgespoints of opposite edges

OctahedronOctahedron

It has eight faces, six vertices and twelve edges It has eight faces, six vertices and twelve edges

Four Four equilateral trianglesequilateral triangles converge in each vertex converge in each vertex

( Euler: 8 + 6 – 12 = 2 )

SymmetriesSymmetries::

intersection of diagonals identify the intersection of diagonals identify the

symmetry centresymmetry centre

3 symmetry axes3 symmetry axes link opposite vertices link opposite vertices

6 symmetry axes6 symmetry axes pass through middle points pass through middle points

of parallel edgesof parallel edges

9 symmetry planes9 symmetry planes, 3 of which pass through , 3 of which pass through

4 parallel edges two by two and 6 passing 4 parallel edges two by two and 6 passing

through 2 opposite vertices and middle through 2 opposite vertices and middle

points of opposite edgespoints of opposite edges

Hexahedron

It has six faces, eight vertices and twelve edgesIt has six faces, eight vertices and twelve edges

Three Three squaressquares converge in each vertex converge in each vertex

( Euler: 6 + 8 – 12 = 2)

Symmetries:Symmetries:

intersection of diagonals identifies the intersection of diagonals identifies the

symmetry centresymmetry centre

9 symmetry axes9 symmetry axes: 3 passing through : 3 passing through

centres of opposite faces, 6 passing through centres of opposite faces, 6 passing through

middle points of opposite edgesmiddle points of opposite edges

9 symmetry planes9 symmetry planes (3 median planes and 6 (3 median planes and 6

diagonal planes)diagonal planes)

IcosahedronIcosahedron

It has twenty faces, twelve vertices and thirty It has twenty faces, twelve vertices and thirty

edgesedges

Five Five equilateral trianglesequilateral triangles converge in each converge in each

vertexvertex

DodecahedronDodecahedron

It has twelve faces, twenty vertices and It has twelve faces, twenty vertices and

thirty edges thirty edges

Three Three pentagonspentagons converge in each vertexconverge in each vertex

(Euler: 20 + 12 – 30 = 2)(Euler: 20 + 12 – 30 = 2)

(Euler: 12 + 20 – 30 = 2)(Euler: 12 + 20 – 30 = 2)

Some symmetries: it has Some symmetries: it has

a a symmetry centersymmetry center,,

axesaxes passing through opposite vertices passing through opposite vertices

of opposite faces, of opposite faces,

planesplanes containing edges of opposite containing edges of opposite

facesfaces

Some symmetries: it hasSome symmetries: it has

a a symmetry centersymmetry center, ,

planesplanes passing through barycenter passing through barycenter

containing one edgecontaining one edge

lineslines passing through opposite vertices of passing through opposite vertices of

opposite facesopposite faces

It is possible to demonstrate that It is possible to demonstrate that

there are only five regular polyhedrathere are only five regular polyhedra

A solid angle must A solid angle must

have at least three have at least three

facesfaces

The sum of the angles The sum of the angles

of the faces must be of the faces must be

less than 360°less than 360°

360°

To construct a polyhedron with To construct a polyhedron with equilateral equilateral triangles:triangles:

3 faces converge at each vertex3 faces converge at each vertex 3 x 60°=180°<360° (tetrahedron) 3 x 60°=180°<360° (tetrahedron)

4 faces converge at each vertex4 faces converge at each vertex4 x 60°=240°<360° (octahedron) 4 x 60°=240°<360° (octahedron)

5 faces converge at each vertex5 faces converge at each vertex5 x 60°=300°<360° (icosahedron)5 x 60°=300°<360° (icosahedron)

It is impossible for 6 or more faces to It is impossible for 6 or more faces to converge in one vertex because:converge in one vertex because:6 x 60° = 360° 6 x 60° = 360°

To construct a polyhedron with To construct a polyhedron with squaressquares::

3 faces converge at each vertex3 faces converge at each vertex

3 x 90°=270°<360° (hexahedron)3 x 90°=270°<360° (hexahedron)

It is impossible for 4 or more faces to converge in one vertex It is impossible for 4 or more faces to converge in one vertex

because: 4 x 90°=360° because: 4 x 90°=360°

To construct a polyhedron with To construct a polyhedron with pentagonspentagons::

3 faces converge at each vertex3 faces converge at each vertex

3 x 108°=324°<360° (dodecahedron)3 x 108°=324°<360° (dodecahedron)

It is impossible for 4 or more faces to converge in one vertex It is impossible for 4 or more faces to converge in one vertex

because: 4 x 108°=432°>360°because: 4 x 108°=432°>360°

An outline of history of PolyhedraAn outline of history of Polyhedra

Hexahedron - earth

Icosahedron - water

Octahedron - air

humidcold

Tetrahedron - fire

hotdry

Leonardo Pisano known as FibonacciLeonardo Pisano known as Fibonacci

““Practica Geometriae”Practica Geometriae”

(1222)(1222)

Piero della FrancescaPiero della Francesca

““De quinque corporibus regularibus” De quinque corporibus regularibus”

(Second half of the 15(Second half of the 15thth century) century)

Luca PacioliLuca Pacioli

““De Divina Proportione” De Divina Proportione”

(1509)(1509)

Leonardo da VinciLeonardo da Vinci

Johannes KeplerJohannes Kepler

““Mysterium Cosmographicum” 1596Mysterium Cosmographicum” 1596

Dual polyhedraDual polyhedra

2020 vertices vertices

3030 edges edges

1212 faces faces

12 vertices12 vertices

3030 edges edges

2020 facesfaces

PP QQdualdual

Dual polyhedraDual polyhedra

66 vertices vertices

1212 edges edges

8 8 faces faces

88 vertices vertices

1212 edges edges

6 6 facesfaces

dualdualPP QQ

Process to convert a polyhedron P to its Process to convert a polyhedron P to its dual Qdual Q

Consider as vertices of Q the centres of the faces of PConsider as vertices of Q the centres of the faces of P

The edges of Q are the segments that connect the The edges of Q are the segments that connect the centres of sequential faces of Pcentres of sequential faces of P

The faces of Q are the polygons that have as vertices the The faces of Q are the polygons that have as vertices the centre of the faces of P centre of the faces of P

QQ

PPPP PP

The prismThe prism

height

base

base

Side face

A A prismprism is a polyhedron bounded by is a polyhedron bounded by two basestwo bases, that are , that are

congruent polygons placed on congruent polygons placed on parallel planesparallel planes,,

and and side facesside faces that are that are parallelogramsparallelograms

The The distancedistance between the between the planesplanes containing the bases is the containing the bases is the

heightheight of the prism of the prism

If the side faces are perpendicular to the planes of the bases, If the side faces are perpendicular to the planes of the bases,

the prism is called a the prism is called a right prismright prism; ; and, if the bases are regular and, if the bases are regular

polygons, the prism is called a polygons, the prism is called a regular prismregular prism

A prism with six rectangular faces is called A prism with six rectangular faces is called rectangular prismrectangular prism

A prism with six faces made by parallelograms is called A prism with six faces made by parallelograms is called

parallelepipedparallelepiped

regular prism rectangular prism parallelepiped

The pyramidThe pyramid

Consider a solid angle with vertex “V” and a plane “Consider a solid angle with vertex “V” and a plane “αα” ”

not passing through “V”not passing through “V”

The part of solid angle containing “V” and delimited by The part of solid angle containing “V” and delimited by

““αα” is called ” is called pyramid pyramid

V vertexV vertex

ABCDEF base ABCDEF base

VAB lateral face (triangle)VAB lateral face (triangle)

VH height (distance vertex V and VH height (distance vertex V and plane plane αα))

VB lateral edgeVB lateral edge

AB edge baseAB edge base

α

V

A B

C

DE

H

M

A pyramid is A pyramid is rightright if its base polygon circumscribes a if its base polygon circumscribes a

circle and the base point of the pyramid height circle and the base point of the pyramid height

corresponds to the centre of the circlecorresponds to the centre of the circle

The The apothemapothem (VM) of a (VM) of a right pyramidright pyramid is the height of one is the height of one

of its facesof its faces

A pyramid is called A pyramid is called regularregular if it is right and the base if it is right and the base

polygon is a regular polygonpolygon is a regular polygon

α

V

regular pyramidregular pyramid

α

right pyramidright pyramid

C

V

Surface area calculusSurface area calculus

The faces of a polyhedron are poligons and we can therefore The faces of a polyhedron are poligons and we can therefore

imagine to open the polyhedron and extend the faces on a planeimagine to open the polyhedron and extend the faces on a plane

The The surface areasurface area of a polyhedron is equal to the sum of the area of a polyhedron is equal to the sum of the area

of all of its facesof all of its faces

The results of the plane figure that we obtain take the name of The results of the plane figure that we obtain take the name of

development planedevelopment plane of the polyhedron of the polyhedron

Volume solidsTwo solids having the same spatial extension or volume Two solids having the same spatial extension or volume

are called equivalentare called equivalent

If two solids can be divided in an equal number of If two solids can be divided in an equal number of

congruent solids, then they are equivalentcongruent solids, then they are equivalent

This is a This is a sufficient but not necessary conditionsufficient but not necessary condition for equivalence for equivalence

between solidsbetween solids

α’

α

If parallel planes intersect two solids so that each If parallel planes intersect two solids so that each

plane defines equivalent sections, then two solids are plane defines equivalent sections, then two solids are

equivalent that is the equivalent that is the volumes volumes of the two solids are of the two solids are

equal equal

This is a This is a sufficient but not necessary conditionsufficient but not necessary condition for equivalence for equivalence

between solidsbetween solids

Cavalieri's PrincipleCavalieri's Principle

)()(// ''' PVPVSS

S S’

P P’

.

For this reason two prisms having equivalent bases and For this reason two prisms having equivalent bases and

congruent height have equal volume:congruent height have equal volume:

VVprismprism =S =Sbb h hTwo pyramids with equivalent bases and congruent heights Two pyramids with equivalent bases and congruent heights

have equal volumehave equal volumeIt is possible to demonstrate that the pyramid’s volume It is possible to demonstrate that the pyramid’s volume

corresponds to a third of the volume of a prism with base and corresponds to a third of the volume of a prism with base and height congruent to those of the pyramid height congruent to those of the pyramid

Therefore the volume of a pyramid isTherefore the volume of a pyramid isVV pyramidpyramid =1/3 S =1/3 Sbb h h

Some of the pictures are taken from Wikipedia

Special thanks to prof. Cinzia Cetraro

for linguistic supervision