1. Kepler-Poinsot Solids
Ciri-Ciri Pepejal Kepler-Poinsot
Minggu 13

2. Bentuk Pentagram
In geometry, a regular star polygon is a self-intersecting, equilateral equiangular polygon.
Bentuk asas ialah pentagon

3. The Pentagram
Satu bucu bagi suatu poligon sekata p sisi (dalam contoh ini ialah pentagon) disambungkan kepada bucu yang tidak bersebelahan (non-adjacent) dan proses diulangi sehingga bucu asal ditemui semula.

4. SIMPLEST REGULAR STAR POLYGON1
Diwakili dengan simbol Schläfli {5/2}.
Sambungkan setiap bucu kedua daripada lima bucu sebuah pentagon 5 2 4 3

5. HEPTAGONAL STAR POLYGON1
Diwakili dengan simbol Schläfli {7/2}.
Sambungkan setiap bucu kedua daripada tujuh bucu sebuah heptagon 7 2 6 3 5 4

6. STELLATION
In1619 Kepler defined stellation for polygons and polyhedra, as the process of extending edges or faces until they meet to form a new polygon or polyhedron.

7. BINA PENTAGRAM E D C B A F J G I H
Pentagram boleh dibina sebagai stellation bagi sebuah pentagon
AB bertemu DC di F
BC bertemu ED di G
AE bertemu CD di H
BA bertemu DE di I
EA bertemu CB di J

8. Kepler’s Stellations
Kepler telah melakukan stellation ke atas dodekahedron
Hasilnya ialah dua daripada polihedra Kepler-Poinsot (regular star polyhedra).

9. POLIHEDRA KEPLER

10. The Small Stellated Dodecahedron

11. The Small Stellated Dodecahedron

12. Membina SS Dodecahedron
Extend the edges of a face of a regular dodecahedron to obtain a pentagram.
Repeat in this way for all twelve faces.
A non-convex polyhedron, the small stellated dodecahedron, is formed.
It has 12 faces (pentagrams), 12 vertices and 30 edges.
Five faces meet in each vertex.
The SS dodecahedron can be constructed by putting appropriated five-sided pyramids upon the faces of a dodecahedron.
Each face of those pyramids is a "golden triangle"

13. The Golden Triangle

14. The Great Stellated Dodecahedron

15. Membina GS Dodecahedron
Consider one of the vertices of a regular icosahedron.
The other endpoints of the five edges starting from the chosen vertex are the vertices of a pentagon.
Extend the sides of this pentagon to obtain a pentagram.
As the icosahedron has 12 vertices,12 pentagrams are formed.
They are the faces of a non-convex polyhedron, the great stellated dodecahedron.
It has 12 faces (pentagrams), 20 vertices and 30 edges.
Three faces meet in each vertex.
The great stellated dodecahedron can be constructed by putting appropriated three-sided pyramids upon the faces of the given icoshedron. Each face of those pyramids is a "golden triangle"
The vertices of the great stellated dodecahedron are also the vertices of a dodecahedron.

16. The Great Stellated Dodecahedron

17. POLIHEDRA POINSOT

18. Great Dodecahedron

19. Membina Great Dodecahedron
Consider one of the vertices of a regular icosahedron.
The other endpoints of the five edges starting from the chosen vertex are the vertices of a pentagon.
As the icosahedron has 12 vertices we obtain 12 pentagons, all situated inside the icosahedron.
They are the faces of a a great dodecahedron.
It has 12 faces (pentagons), 12 vertices and 30 edges.
Three faces meet in each vertex.
The great dodecahedron has the same vertices and edges as the original icosahedron.

20. Great Dodecahedron

21. Great Icosahedron

22. Membina Great Icosahedron
Connect each vertex with its five "nearest neighbours" we obtain the edges of the icosahedron.
Each vertex also has five "next nearest neighbours".
Five combinations of the vertex chosen and two of its "next nearest neighbours" form equilateral triangles inside the icosahedron.

23. Membina Great Icosahedron
Each vertex belongs to five triangles and there are 20 equilateral triangles.
They are the faces of a regular non-convex polyhedron, called a great icosahedron.
It has 20 faces (triangles), 12 vertices and 30 edges. Five faces meet in each vertex.

24. Great Icosahedron

25. Small Stellated Dodecahedron
Great Stellated Dodecahedron
Great Icosahedron
Great Dodecahedron

26. RUMUSAN Name: Dual Faces Verticies Edges Small Stellated Dodecahedron
Great Dodecahedron 12 30
Great Stellated Dodecahedron
Great Icosahedron 20
Great Dodecahedron