# kerja kursus polyhedral

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Bahagian 2 Dalam kumpulan dua orang, anda dikehendaki membina model yang terdiri daripada dua jenis polyhedral.

Polyhedron

The word polyhedron has slightly different meanings in geometry and algebraic geometry. In geometry, a polyhedron is simply a three-dimensional solid which consists of a collection of polygons, usually joined at their edges. The word derives from the Greek poly (many) plus the Indo-European hedron (seat). A polyhedron is the three-dimensional version of the more general polytope (in the geometric sense), which can be defined in arbitrary dimension. The plural of polyhedron is "polyhedra" (or sometimes "polyhedrons"). The term "polyhedron" is used somewhat differently in algebraic topology, where it is defined as a space that can be built from such "building blocks" as line segments, triangles, tetrahedra, and their higher dimensional analogs by "gluing them together" along their faces (Munkres 1993, p. 2). More specifically, it can be defined as the underlying space of a simplicial complex (with the additional constraint sometimes imposed that the complex be finite; Munkres 1993, p. 9). In the usual definition, a polyhedron can be viewed as an intersection of half-spaces, while a polytope is a bounded polyhedron.

1

A convex polyhedron can be formally defined as the set of solutions to a system of linear inequalities

where

is a real

matrix and is a real -vector. Although usage varies, most authors

additionally require that a solution be bounded for it to define a convex polyhedron. An example of a convex polyhedron is illustrated above. The following table lists the name given to a polyhedron having faces for small . When used without qualification for polyhedron for which symmetric forms exist, the term may mean this particular polyhedron or may mean a arbitrary -faced polyhedron, depending on context.

polyhedron 4 5 6 7 8 tetrahedron pentahedron hexahedron heptahedron octahedron2

9

nonahedron

10 decahedron 11 undecahedron 12 dodecahedron 14 tetradecahedron 20 icosahedron 24 icositetrahedron 30 triacontahedron 32 icosidodecahedron 60 hexecontahedron 90 enneacontahedron

A polyhedron is said to be regular if its faces and vertex figures are regular (not necessarily convex) polygons (Coxeter 1973, p. 16). Using this definition, there are a total of nine regular polyhedra, five being the convex Platonic solids and four being the concave (stellated) Kepler-Poinsot solids. However, the term "regular polyhedra" is sometimes used to refer exclusively to the Platonic solids (Cromwell 1997, p. 53). The dual polyhedra of the Platonic solids are not new polyhedra, but are themselves Platonic solids. A convex polyhedron is called semiregular if its faces have a similar arrangement of nonintersecting regular planar convex polygons of two or more different types about each polyhedron vertex (Holden 1991, p. 41). These solids are more commonly called the Archimedean solids, and there are 13 of them. The dual polyhedra of the Archimedean solids are 13 new (and beautiful) solids, sometimes called the Catalan solids. A quasiregular polyhedron is the solid region interior to two dual regular polyhedra (Coxeter 1973, pp. 17-20). There are only two convex quasiregular polyhedra: the cuboctahedron3

and icosidodecahedron. There are also infinite families of prisms and antiprisms. There exist exactly 92 convex polyhedra with regular polygonal faces (and not necessarily equivalent vertices). They are known as the Johnson solids. Polyhedra with identical polyhedron vertices related by a symmetry operation are known as uniform polyhedra. There are 75 such polyhedra in which only two faces may meet at an polyhedron edge, and 76 in which any even number of faces may meet. Of these, 37 were discovered by Badoureau in 1881 and 12 by Coxeter and Miller ca. 1930. Polyhedra can be superposed on each other (with the sides allowed to pass through each other) to yield additional polyhedron compounds. Those made from regular polyhedra have symmetries which are especially aesthetically pleasing. The graphs corresponding to polyhedra skeletons are called Schlegel graphs.

SAMPEL POLYHEDRAL 1.

4

FIG.3 - The truncated icosido-decahedron (on the left) and the rhombic icosi-dodecahedron (on the right) are a couple of Archimedean polyhedra generated from the intersection of three polyhedra: an icosahedron, a dodecahedron and a rhombic triacontahedron. When the ratios between the distances of their faces from the centre of the resulting polyhedron get appropriate values, the faces are regular polygons: together with squares, in the first case there are decagons and hexagons that, in the second case, become pentagons and triangles, respectively.

2. Model Polyhedral

5

3.MODEL POLYHEDRAL

4.MODEL POLYHEDRAL

5. MODEL POLYHEDRAL6

6. MODEL POLYHEDRAL

6. MODEL POLYHEDRAL

POLYHEDRAL

YANG

DIBUAT

OLEH7

KUMPULAN

KAMI

IALAH

GABUNGAN PIRAMID DAN KUBUS LANGKAH-LANGKAH MEMBUAT: Langkah 1: Menyediakan kertas berbentuk segiempat dan membuta libatan

Langkah 2 : Membuat 5 lipatan yang sama pada kertas warna yang lain.

Langkah 3 : Cantumkan keenam-enam lipatan tersebut untuk menjadi polyhedral yang berwarna-warni

REFLEKSI:9

Pertama sekali, kami ingin mengucapkan ribuan terima kasih kepada pensyarah En. Chiong untuk mengarahkan tajuk polyhedral ini untuk kami. Kami berasa bangga juga kerana berpeluang lagi bekerjasama untuk

BIBLIOGRAFI10

Rujukan daripada laman web: 1. http://mathworld.wolfram.com/Polyhedron.html 2. http://flickrhivemind.net/Tags/polyhedron 3. www.boxvox.net/2008/05/polyhedral-mode.html 4. www.mi.sanu.ac.rs/vismath/zefirocorrection/__Zefiro-Ardigo'_icosahedral_ polyhedra_updating.htm 5. http://numb3rs.wolfram.com/406/ 6. http://www.platonicsolids.info/origami.htm TEACHING

7. http://www.maa.org/mathland/mathtrek_04_23_06.html 8. http://en.wikipedia.org/wiki/File:Uniform_polyhedron-53-s012.png 9. http://www.123rf.com/photo_6761776_polyhedral-figure-of-a-star-with-grad ient-vector-3d.html

Borang Rekod Kolaborasi Kerja Kursus11

NAMA PELAJAR NO. MATRIK KP KUMPULAN MATA PELAJARAN

: : : :

TIONG ING CHIONG / HII ING YEU 830601-13-5545 / 840927-13-5624 PPG PJK AMBILAN JUN 2011 WAJ3105 LITERASI NOMBOR : EN. CHIONG YEW KAI Nama /Tandatangan daripada TIONG ING CHIONG

PENSYARAH PEMBIMBING Tarikh 2.7.2011

Aktiviti yang dijalankan Menerima pensyarah. Penerangan daripada pensyarah. soalan tugasan

HII ING YEU 15.7.2011 Membahagikan tugas Mencari maklumat dari TIONG ING CHIONG

internet/perpustakaan awam. HII ING YEU 27.8.2011 Berbincang bersama untuk TIONG ING CHIONG