Download presentation

Presentation is loading. Please wait.

Published byAlejandro Lewis Modified over 2 years ago

1
Liceo Scientifico Isaac Newton Maths coursePolyhedra Professor Tiziana De Santis

2
facevertex edge diagonal A convex polyhedron is the part of space bounded by n polygons (with n 4) belonging to different planes, so that each edge of the polyhedron is the intersection of two of them

3
Eulers relation All convex polyhedra satisfy this important relation between the numbers of faces (F), of vertices (V) and of edges (E) – 90 = 2 F + V – E = 2 32 faces edges 60 vertices

4
Regular polyhedra A polyhedron is said to be regular if its faces are regular and congruent polygons, and its dihedral angles and solid angles are also congruent These solids are also called platonic

5
Tetrahedron It has four faces, four vertices and six edges Three equilateral triangles converge in each vertex ( Euler: – 6 = 2 ) Symmetries: - 6 planes passing through the barycentre containing one edge - 3 lines passing through middle points of opposite edges

6
Octahedron It has eight faces, six vertices and twelve edges Four equilateral triangles converge in each vertex ( Euler: – 12 = 2 ) Symmetries: intersection of diagonals identify the symmetry centre 3 symmetry axes link opposite vertices 6 symmetry axes pass through middle points of parallel edges 9 symmetry planes, 3 of which pass through 4 parallel edges two by two and 6 passing through 2 opposite vertices and middle points of opposite edges

7
Hexahedron It has six faces, eight vertices and twelve edges Three squares converge in each vertex ( Euler: – 12 = 2) Symmetries: intersection of diagonals identifies the symmetry centre intersection of diagonals identifies the symmetry centre 9 symmetry axes: 3 passing through centres of opposite faces, 6 passing through middle points of opposite edges 9 symmetry planes (3 median planes and 6 diagonal planes)

8
Icosahedron It has twenty faces, twelve vertices and thirty edges Five equilateral triangles converge in each vertex Dodecahedron It has twelve faces, twenty vertices and thirty edges Three pentagons converge in each vertex (Euler: – 30 = 2) (Euler: – 30 = 2) Some symmetries: it has a symmetry center, axes passing through opposite vertices of opposite faces, planes containing edges of opposite faces Some symmetries: it has a symmetry center, planes passing through barycenter containing one edge lines passing through opposite vertices of opposite faces

9
It is possible to demonstrate that there are only five regular polyhedra A solid angle must have at least three faces The sum of the angles of the faces must be less than 360° 360°

10
To construct a polyhedron with equilateral triangles: 3 faces converge at each vertex 3 x 60°=180°<360° (tetrahedron) 3 x 60°=180°<360° (tetrahedron) 4 faces converge at each vertex 4 x 60°=240°<360° (octahedron) 5 faces converge at each vertex 5 x 60°=300°<360° (icosahedron) It is impossible for 6 or more faces to converge in one vertex because: 6 x 60° = 360°

11
To construct a polyhedron with squares: 3 faces converge at each vertex 3 x 90°=270°<360° (hexahedron) It is impossible for 4 or more faces to converge in one vertex because: 4 x 90°=360° To construct a polyhedron with pentagons: 3 faces converge at each vertex 3 x 108°=324°<360° (dodecahedron) It is impossible for 4 or more faces to converge in one vertex because: 4 x 108°=432°>360°

12
An outline of history of Polyhedra Hexahedron - earth Icosahedron - water Octahedron - air humid cold Tetrahedron - fire hot dry

13
Leonardo Pisano known as Fibonacci Practica Geometriae (1222) Piero della Francesca De quinque corporibus regularibus (Second half of the 15 th century) Luca Pacioli De Divina Proportione (1509)

14
Leonardo da Vinci Johannes Kepler Mysterium Cosmographicum 1596

15
Dual polyhedra 20 vertices 30 edges 12 faces 12 vertices 30 edges 20 faces P Q dual

16
Dual polyhedra 6 vertices 12 edges 8 faces 8 vertices 12 edges 6 faces dual P Q

17
Process to convert a polyhedron P to its dual Q Process to convert a polyhedron P to its dual Q Consider as vertices of Q the centres of the faces of P The edges of Q are the segments that connect the centres of sequential faces of P The faces of Q are the polygons that have as vertices the centre of the faces of P Q P P P

18
The prism height base Side face A prism is a polyhedron bounded by two bases, that are congruent polygons placed on parallel planes, and side faces that are parallelograms The distance between the planes containing the bases is the height of the prism

19
If the side faces are perpendicular to the planes of the bases, the prism is called a right prism; and, if the bases are regular polygons, the prism is called a regular prism A prism with six rectangular faces is called rectangular prism A prism with six faces made by parallelograms is called parallelepiped regular prism rectangular prism parallelepiped

20
The pyramid Consider a solid angle with vertex V and a plane α not passing through V The part of solid angle containing V and delimited byα is called pyramid V vertex ABCDEF base VAB lateral face (triangle) VH height (distance vertex V and plane α) VB lateral edge AB edge base α V A B C D E H

21
M A pyramid is right if its base polygon circumscribes a circle and the base point of the pyramid height corresponds to the centre of the circle The apothem (VM) of a right pyramid is the height of one of its faces A pyramid is called regular if it is right and the base polygon is a regular polygon α V regular pyramid α right pyramid C V

22
Surface area calculus The faces of a polyhedron are poligons and we can therefore imagine to open the polyhedron and extend the faces on a plane The surface area of a polyhedron is equal to the sum of the area of all of its faces The results of the plane figure that we obtain take the name of development plane of the polyhedron

23
Volume solids Two solids having the same spatial extension or volume are called equivalent If two solids can be divided in an equal number of congruent solids, then they are equivalent This is a sufficient but not necessary condition for equivalence between solids

24
α α If parallel planes intersect two solids so that each plane defines equivalent sections, then two solids are equivalent that is the volumes of the two solids are equal This is a sufficient but not necessary condition for equivalence between solids Cavalieri's Principle S S P P.

25
For this reason two prisms having equivalent bases and congruent height have equal volume: V prism =S b h Two pyramids with equivalent bases and congruent heights have equal volume It is possible to demonstrate that the pyramids volume corresponds to a third of the volume of a prism with base and height congruent to those of the pyramid Therefore the volume of a pyramid is V pyramid =1/3 S b h

26
Some of the pictures are taken from Wikipedia Special thanks to prof. Cinzia Cetraro for linguistic supervision

Similar presentations

© 2016 SlidePlayer.com Inc.

All rights reserved.

Ads by Google