Liceo Scientifico Isaac Newton Maths course Polyhedra Professor Tiziana De Santis
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Transcript of Liceo Scientifico Isaac Newton Maths course Polyhedra 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
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