Vol. 2,No. 1 TyabandhaJournalofArtsandScience First Year Report Kittisak Nui Tiyapan (cid:0) Introduction Geometry and mathematics The simplest geometrical (cid:2)gure is a circle while the simplest of all polygons is a triangle. The degree of freedom of triangles increases fromtheequilateraltotheisoscelesandtherighttrianglestothescalene triangles. While spending the summer of 1990 in a traineeship through AIESEC I was introduced to a geometrical puzzle which, as I came to learn later, is called the (cid:3)exatube, but which I conjectured at the time that was from ancient China. This puzzle is made up of sixteen right isosceles triangles tiled into four squares, each comprising of four triangles, which are in turn joined together to form a loop. It can be easily made up using some hard papers, a pair of scissors and cello tape. There are in total twenty hinges, four of which are as long as the hypotenuse while the other sixteen have their length equal to the shorter side of the triangle. By turning these rigid triangles upon their hinges the inside surface of the strip can become the outside and vice versa. I found one solution a week later and back in Thailand a friend of mine found another solution. I think that these two are the only possible solutions but have not been able to proof it. Having this interest in geometrical puzzles I was delighted at the time to (cid:2)nd that the symbol for the men’s toilets in Budapest is an equilateral triangle, while that for women’s toilets is a circle. No other things beside these are written. Obviously these two geometrical forms suf(cid:2)ce and are fully understood by the whole population. In general dimension we talk about spheres. A sphere in dimen› (cid:1) sions has its volume, , proportional to and its surface, , to 1, (cid:2) (cid:3)(cid:5)(cid:4) (cid:6) (cid:3)(cid:7)(cid:4)(cid:9)(cid:8) so that 1 . (cid:18) (cid:2)(cid:11)(cid:10)(cid:12)(cid:6) (cid:4)(cid:14)(cid:13)(cid:16)(cid:15)(cid:17)(cid:4)(cid:9)(cid:8) The area of sphere in three dimensions is 4 2 and the vol› (cid:6)(cid:20)(cid:19) (cid:21)(cid:22)(cid:3) ume 4 3. When 1, 3 1 3 22 3, 3 1 3 22 3, (cid:2)(cid:23)(cid:19) 3(cid:21)(cid:22)(cid:3) (cid:2)(cid:23)(cid:19) (cid:3)(cid:24)(cid:19)(cid:23)(cid:25)(cid:27)(cid:26)(cid:28)(cid:25) (cid:29)(cid:30)(cid:21) (cid:31) (cid:13) (cid:29) (cid:13) (cid:26) (cid:29)(cid:30)(cid:21) (cid:31) (cid:13) (cid:29) (cid:13) or 1 2 3 3 1 3 22 3 which are numerically 0310175 0537239 , (cid:26)!(cid:25) (cid:31) (cid:13) (cid:26) (cid:29)"(cid:21) (cid:31) (cid:13) (cid:29) (cid:13) (cid:25) # (cid:25) # $ 0.62035, 0310175, 0310175 0537239 in that order. Therefore (cid:25) # (cid:25) # % # $ &’(cid:19) 62 3-3 483598 1 (cid:6))(*,+ (cid:19) (cid:13) (cid:21).(cid:19) # (cid:0) Written while the author wasat Chemical Engineering, UMIST,Su› pervisor, Professor Graham Arthur Davies, The University of Manch› ester, 7 October 2002 /10 VaenSryayudhya,Editor January2005 5 TyabandhaJournalofArtsandScience Vol. 2,No. 1 When 8, 6 1 3 and therefore 462 3 1 3 193439. In (cid:2) (cid:19) (cid:3) (cid:19) (cid:26) (cid:29)(cid:30)(cid:21),(cid:31) (cid:13) (cid:6) (cid:19) (cid:13) (cid:21) (cid:13) (cid:19) # order to (cid:2)nd , the surface area(cid:1)per unit volume, one divide the area & by 2 3, in other words 1 3 2 3. (cid:2) (cid:13) & (cid:19) (cid:2) (cid:13) (cid:6) (cid:29)(cid:7)(cid:2) (cid:19) (cid:6) (cid:29)(cid:7)(cid:2) (cid:13) (cid:2) The same is true for other polyhedra. For example in atetrahedron where is the length of the side, the vertices can be (cid:6) (cid:6) (cid:2) (cid:2) (cid:2) (cid:2) (cid:2) (cid:2) (cid:3) (cid:3) (cid:4)(cid:3) (cid:3) (cid:3) (cid:5)(cid:3) (cid:3) - 3 (cid:3) (cid:7)(cid:8)(cid:3) (cid:3) - 3 (cid:3) 1(cid:9) 93 (cid:7) 0 0 0 0 0 0 (cid:26) (cid:31) (cid:30)(cid:26) (cid:31) 2 2 2 6 2 35 (cid:2) When 1, one can obtain by solving the equation (cid:2) (cid:19) (cid:13) (cid:13)(cid:17)(cid:5)(cid:18) (cid:13) (cid:13) (cid:18) (cid:13) (cid:13) (cid:18) (cid:13) (cid:2)0 0 0 1 (cid:13) 1 (cid:10)(cid:11)(cid:11) (cid:13)(cid:13)(cid:13) (cid:14) (cid:15) 0(cid:2) 0 1 (cid:13)(cid:13)(cid:13)(cid:19) 1 (cid:19) 6abs (cid:11)(cid:12) (cid:13)(cid:13) (cid:14)2 23(cid:2) 0 (cid:2) 1(cid:13)(cid:13) # (cid:15) (cid:16) 3 1 93 1 2 6 2 35 (cid:2) (cid:20) T(cid:2) his gives 2 35(cid:21) 16 20409 as the only real positive answer. When (cid:19) 31 (cid:19) # is doubled, increases from 1 to 8, which means that one would be dividing (cid:6) by(cid:2) (cid:2) 23 to obtain & . (cid:22) (cid:24)(cid:23) (cid:26)(cid:25) The perim(cid:24)(cid:23)et(cid:28)er(cid:27) of a tri(cid:28)an(cid:27)gle is 3 and th(cid:22) e area - 3 4 2. (cid:19) (cid:1) (cid:6) (cid:19) (cid:26) (cid:29) (cid:31) When 1, 2 31 4 151967. Therefore 455901. (cid:6) (cid:19) (cid:1) (cid:19) (cid:29) (cid:13) (cid:19) # (cid:19) # Verticessharedbytwocellsmakeupacommonfacebetweenthem. Two way have been tried for (cid:2)nding the edges. The (cid:2)rst one was by looking at all neighbouring cells of every cell in turn three at a time. The edges are then made up of those vertices that are common among these three cells. Only those edges which have exactly two vertices are considered. They are called goodedges as contrasted with edges on the boundary. This is a much longer way than the second one, which is to consider vertices common to any two faces of a cell. Similar to the (cid:2)rst case, such vertices forms a good edge if and only if there are only two of them. The two methods above give exactly the same list of edges, so they con(cid:2)rm each other. It has been tested that all edges having more than two vertices are boundary ones, that is they have at least one vertex outside the boundary of the unit cube considered. By drawing some of the cel(cid:29)(cid:31)ls(cid:30)! #a"%s$(cid:24)a&(’(cid:24)s’olid using (cid:2)ll command it has been tested that the result from covers the entire cell surface. This con(cid:2)rms the step where areas are calculated. The hexagon or honeycomb is perhaps the pattern which is most frequently found in nature. Even though the world we live in is three› dimensional, cellsnormally divideandspreadin two dimensions in the form of layers. Moreover, they are packed in these layers in patterns which most often resemble the honeycomb cf Williams and Bjerknes, (cid:26) 1972 . (cid:31) 6 January2005 VaenSryayudhya,Editor Vol. 2,No. 1 TyabandhaJournalofArtsandScience An octagon is an eight›sided polygon. It is the shape of the cross section of every chimney in the mills built in Manchester during its industrial era of the nineteenth century, as well as that of the turrets in the Main Building of UMIST. Perhaps one of the reasons for its pop› ularity is that it looks strong while having the style of a good taste. May be the reason why it looks strong is that it possesses eight axes of symmetry, on top of another symmetry around the origin. There are nine regular polyhedra. Among these are (cid:2)ve regular convex solids known to the ancient Greek called Platonic polyhedra. They are tetrahedron, cube, dodecahedron, octahedron, and icosahe› dron. They have regular congruent faces and regular polyhedral angle vertices. Their face angles and their dihedral angles at every vertex are equal. The other four regular polyhedra have only been discovered muchlaterandarenotconvex. TheyarecalledtheKepler›Poinsotpoly› hedra and are nonconvex. The small stellated dodecahedron and the great stellated dodecahedron were found by Kepler 1571(cid:150)1630 . The (cid:26) (cid:31) great icosahedron and the great dodecahedron were found by Poinsot 1777(cid:150)1859 . The small stellated dodecahedron and the great dodeca› (cid:26) (cid:31) hedron do not satisfy Euler’s equation. The process of creating it by extending nonadjacent faces until they meet is called stellating. There are also polyhedra called quasi›regular. The semi›regular polyhedra are called the Archimedean polyhe› dra. Here all faces are regular polygons but not all are of the same kind. Every vertex is congruent to all others. They comprise of an in(cid:2)nite group of prisms, an in(cid:2)nite group of antiprisms or prismoid, and another thirteen polyhedra. Each prism or prismoid is made up of two regular polygons on parallel planes where the vertices are aligned in the former case or shifted half way to the next neighbouring vertices in the latter case. Each vertex in prisms is joined to a corresponding vertex of the opposite polygon, while in prismoid it is joined to two corresponding vertices. All faces of an Archimedean solid are regular and all its polyhedral angle vertices congruent. On the other hand the Archimedean duals have the property that all their faces are congruent to one another and all their polyhedral angles regular. These solids are important in crystallography. They are vertically regular and include an in(cid:2)nite group of dipyramids, an in(cid:2)› nite group of trapezohedra, and additionally thirteen other polyhedra. Thesurfaceareaperunitvolume ofasolidcanbecomputedfrom & the actual volume and the actual surface area as 1 3 (cid:2) (cid:6) &(cid:12)(cid:19) (cid:2) (cid:13) (cid:6) (cid:29)(cid:7)(cid:2) (cid:19) 2 3. (cid:6) (cid:2) (cid:8) (cid:13) VaenSryayudhya,Editor January2005 7 TyabandhaJournalofArtsandScience Vol. 2,No. 1 (cid:23) (cid:22) Polygon numerical (cid:1) (cid:0) (cid:6) (cid:26) (cid:31) Triangle 3 - 3 4 2 4.55901 (cid:26) (cid:29) (cid:31) (cid:1) Square 4 2 4 (cid:1) Pentagon 5 5 1 - 5 2 4 10 2- 5 1 2 3.8119 (cid:4) (cid:26) % (cid:31) (cid:1) (cid:29)(cid:3)(cid:2) (cid:26) (cid:25) (cid:31) (cid:13) Hexagon 6 3- 3 2 2 3.7224 (cid:1) (cid:29) Heptagon 7 7 4 2tan 5 14 3.6721 (cid:26) (cid:29) (cid:31)!(cid:1) (cid:26) (cid:21) (cid:29) (cid:31) Octagon 8 2 2tan 3 8 3.6407 (cid:1) (cid:26) (cid:21) (cid:29) (cid:31) Nonagon 9 9 4 2tan 7 18 3.6198 (cid:26) (cid:29) (cid:31) (cid:1) (cid:26) (cid:21) (cid:29) (cid:31) 1 2 Decagon 10 5 5 - 5 2(cid:4) (cid:13) 2 1 - 5 0.3605 (cid:2) (cid:26) % (cid:31) (cid:29) (cid:1) (cid:29) (cid:26) (cid:25) % (cid:31) Undecagon 11 11 4 2tan 9 22 3.5944 (cid:26) (cid:29) (cid:31) (cid:1) (cid:26) (cid:21) (cid:29) (cid:31) Dodecagon 12 3 2 - 3 2 3.5863 (cid:26) % (cid:31) (cid:1) Table 1 Perimeterperunitareaof ›gons. (cid:0) The tetrahedron is self›dual. The octahedron is dual to the cube while the dodecahedron the icosahedron. The nearest neighbour and minimum spanning tree have been ap› plied to the problem of taxonomy in botany. Clayton 1972 , work› (cid:26) (cid:31) ing on the characters of plants to manually classify them eg Clayton, (cid:26) 1970 with the use of only the binary dendrogram and trial and error, (cid:31) adopted a numerical method which (cid:2)nds the minimum spanning tree in a multi›dimensional character space. Since taxonomy can be consid› ered as a kind of dictionary, it is possible to apply a similar approach to machine translation and the compilation of dictionaries. The icosahedron is common shape found among viruses. The polyhedra from Figure 1 to 3 are semi›regular. a b (cid:26) (cid:31) (cid:26) (cid:31) Figure 1 a Truncated tetrahedron, triakistetrahedron, 2 33. b Octa› (cid:26) (cid:31) ( (cid:26) (cid:31) hemioctahedron,octahemioctacron,3 233 (cid:29) ( 8 January2005 VaenSryayudhya,Editor Vol. 2,No. 1 TyabandhaJournalofArtsandScience a b (cid:26) (cid:31) (cid:26) (cid:31) Figure 2 a Tetrahemihexahedron,tetrahemihexacron, 3 2 32. b Trun› (cid:26) (cid:31) (cid:29) ( (cid:26) (cid:31) catedoctahedron,tetrakishexahedron,243 ( a b (cid:26) (cid:31) (cid:26) (cid:31) Figure 3 a Truncatedcube,triakisoctrahedron,2 34. b Rhombicubocta› (cid:26) (cid:31) ( (cid:26) (cid:31) hedron,deltoidalicositetrahedron,342 ( Polyhedra in Figure 4 are snub polyhedra. VaenSryayudhya,Editor January2005 9 TyabandhaJournalofArtsandScience Vol. 2,No. 1 a b (cid:26) (cid:31) (cid:26) (cid:31) Figure 4 a Pentagrammic crossed antiprism, pentagrammic concave del› (cid:26) (cid:31) tohedron, 225 3. b Pentagrammicantiprism,pentagrammicdeltohedron, ( (cid:29) (cid:26) (cid:31) 225 2 ( (cid:29) The surface of Fullerine is made up of pentagons six›sided (cid:2)g› ures. Its shape represents that of the geodesic domes developed by Buckminster Fuller, and hence the name Fullerine. The latter may ei› ther be hexagons or (cid:2)gures all the six sides in each one of which form two sets of three sides having an equal length. The simplest Fuller› ine, the carbon›60 molecule, has the same shape as that of a football and a handball. With some thought the reason for this is not dif(cid:2)cult to see. With its thirty›two faces it closely resemble the sphere. Also the two different shapes of all its components are symmetrically dis› tributed andthereforeenable colouring with only two differentcolours, namely one for each of the two shapes. To see how this helps, suppose one made a football in the shape of a bloated dodecahedron. Then it would be impossible to colour it using more than one colour at the same time of giving it a symmetrical appearance when viewed from more than a few directions. With the Fullerine shape and the colour› ing scheme mentioned, however, the football looks symmetrical when viewed from 54 different directions symmetrically distributed around it. These directions corresponds to those when one looks at it in the direction perpendicular to the centre of each of its faces and when in the direction through the middle of each of the 22 edges lying between two hexagonal faces. Making polyhedron models is an educating experience. Contrary to the general believe that you need to make an accurate drawing for the required parts Wenninger, 1971 , this needs not be so. Examples (cid:26) (cid:31) 10 January2005 VaenSryayudhya,Editor Vol. 2,No. 1 TyabandhaJournalofArtsandScience of this arethe origami models of polyhedra where complex polyhedron structures are made from interlocking pieces each of which is made by folding a piece of paper of a rectangular or square shape. A set of elements with the sum and the product of any two el› ements de(cid:2)ned is a commutative ring if under these two operations it satis(cid:2)es the following postulates: closure, uniqueness, commutative, associative, and distributive laws, identity zero and unity , and addi› (cid:26) (cid:31) tive inverse. An integral domain is an ordered domain if its positive elements satisfy the laws of addition, multiplication, and trichotomy. A subset of an ordered domain is well›ordered if every nonempty subset of it contains a smallest member. (cid:1) means that .(cid:1) is divisi› (cid:0)(cid:22)( ble by . (cid:3)The Euclidean algorithm or division algorithm states that (cid:0) (cid:1)(cid:3)(cid:2) 0 (cid:1) .(cid:27) Two integers are relatively prime if their only c(cid:0)’o(cid:19)mmon% d(cid:3) ivis(cid:4) or(cid:3)(cid:6)s (cid:5)are 1. (cid:1) mod if and only if (cid:1) . The (cid:0)(cid:8)(cid:7) (cid:26) (cid:9) (cid:31) (cid:9) ((cid:17)(cid:26)(cid:10)(cid:0) (cid:25) (cid:31) commutative ring is the properties of multiplication and addition of (cid:11) 2 even 0 and odd 1 numbers. (cid:1) (cid:26) (cid:31) (cid:26) (cid:31) 0 1 0 1 % 0 0 1 0 0 0 1 1 0 1 0 1 The following is . (cid:1) (cid:11) 5 0 1 2 3 4 0 1 2 3 4 % 0 0 1 2 3 4 0 0 0 0 0 0 1 1 2 3 4 0 1 0 1 2 3 4 2 2 3 4 0 1 2 0 2 4 1 3 3 3 4 0 1 2 3 0 3 1 4 2 4 4 0 1 2 3 4 0 4 3 2 1 There is a close link between geometry and algebra. Geometri› cal surfaces can be described as algebraical equations. For example, for circles and polygons the equations are binary quadratic, while for spheres and polyhedra they are ternary quadratic. Even one›sided sur› faces can be described algebraically. The equation of Klein bottle, when deformed into a sphere with two circles removed and replaced by two cross›caps, is a quartic equation (cid:2) (cid:2) (cid:2) (cid:4)(cid:3) 2 2 2 (cid:1) 2 2 2 2 2 2 (cid:1) 2 2 (cid:0) (cid:26) %(cid:13)(cid:12) (cid:31) (cid:26) (cid:25) (cid:25)(cid:14)(cid:12) (cid:31) (cid:19)(cid:16)(cid:15) (cid:26)(cid:10)(cid:0) % (cid:12) (cid:31) while the Steiner surface is also a quartic one (cid:2) (cid:2) (cid:2) 2 2 2 2 2 2 0 (cid:12) (cid:15) %(cid:17)(cid:15) % (cid:12) % (cid:12)(cid:18)(cid:15))(cid:19) # Two surfaces is homomorphic to each other if it is possible to con› tinuously transformoneinto theother. Allconvexpolyhedraarehomo› morphic to a sphere. The Steiner surface is homomorphic to the hepta› hedron, which is an Archimedean polyhedron with diametral plane. VaenSryayudhya,Editor January2005 11 TyabandhaJournalofArtsandScience Vol. 2,No. 1 In the plane, a second›degree equation gives either two straight lines, a circle, an ellipse, a parabola, or a hyperbola. In space, it can give two planes, cylinders and cones circular, elliptic, parabolic, or (cid:26) hyperbolic , sphere, spheroid, ellipsoid, two hyperboloids, and elliptic (cid:31) (cid:26) or hyperbolic paraboloid. (cid:31) Partition, tessellation and division of space are the same thing. In the context of set theory, a partition of set is a family of sets , , , 1 2 (cid:0) (cid:6) (cid:6) #(cid:9)# # which are subsets of , such that ; ; , (cid:6)(cid:2)(cid:1) (cid:0) (cid:6)(cid:2)(cid:3)(cid:5)(cid:19)(cid:7)(cid:4) (cid:6) (cid:6)(cid:8)(cid:3)(cid:10)(cid:9) (cid:6)(cid:12)(cid:11) (cid:19)(cid:13)(cid:6) (cid:14) (cid:3) (cid:6)(cid:8)(cid:3) (cid:19)(cid:15)(cid:0) where , 1, 2, , and . cf Berge, 1958 A further condition (cid:16) (cid:17) (cid:19) # # # (cid:18) (cid:16)(cid:19)(cid:19)(cid:20)(cid:4) (cid:17) (cid:26) (cid:31) that makes any tessellation a Voronoi one is that, for all there exists (cid:16) a unique point within such that every point in is closer to (cid:0)(cid:21)(cid:3) (cid:6)(cid:8)(cid:3) (cid:6)(cid:2)(cid:3) (cid:0)(cid:22)(cid:3) than to any other , . (cid:0)(cid:23)(cid:11) (cid:17)(cid:24)(cid:19)(cid:25)(cid:4) (cid:16) Voronoi tessellation in three dimensions can be constructed by imagining each region as a spherical cell growing outwards to meet neighbouring cells and continue growing to (cid:2)ll the gaps. The centre of each sphere is a unique nucleus point of the region such that it is closest to any point belonging to that region than any nuclei points. If the rate of growth is the same from every cell, the resulting partitions will be planes which can be described by ternary quadratic equations. However, if this rate differs from one cell to another, the partitions will be curved surfaces and the result is a non›Voronoi tessellation. It is possible to impose a constraint of minimum distance between neigh› bouring nuclei. Such cases can be looked at as spheres of an equal nonzero radius expanding away from nucleus centre points. If the radii differ from one sphere to another, or if some nonspherical solids are used instead of spheres, the tessellation obtained will be non›Voronoi. Consider the case where all spheres are of equal size. If these spheres already touch their neighbours before the expanding starts, the case is that of packed spheres expanded to form a Voronoi tessella› tion. There are two types of close›packing: cubic face›centred and (cid:26) (cid:31) hexagonal. In both cases each sphere has twelve neighbours. Both cases have the same density, which is (cid:15) . The Voronoi regions pro› 3(cid:26) 2 duced from the cubic case are rhombic dodecahedra and the faces are rhombuses. In the case of hexagonal close›packing, the correspond› ing regions are trapezo›rhombic dodecahedra and the faces are either rhombics or trapezia. Where the spheres meet with their three neigh› bours in the layer above and their three neighbours in the layer below, the faces are rhombics. Where they meet with the six neighbours on the same layer they are trapezia. For geometrical calculation, an example of a de(cid:2)nitive book is that written by Salmon 1912 . (cid:26) (cid:31) The gamma function, (cid:3) (cid:3) (cid:27)(cid:29)(cid:28) Γ 1d Re 0 1 (cid:26)(cid:10)(cid:15) (cid:31) (cid:19) (cid:8) / (cid:31)"! (cid:8) (cid:31) (cid:26)(cid:10)(cid:15) (cid:31)$# (cid:26) (cid:31) 0 (cid:30) 12 January2005 VaenSryayudhya,Editor Vol. 2,No. 1 TyabandhaJournalofArtsandScience got its name from Legendre and is known as the Euler gamma function or simply the second Euler function. The formula Γ 1 Γ ! (cid:26)(cid:10)(cid:15) % (cid:31) (cid:19) (cid:15) (cid:26) (cid:15) (cid:31) (cid:19) (cid:15) recursively calculates the gamma function from, for instance, Γ 1 5 (cid:26) (cid:29) (cid:31)(cid:1)(cid:0) 45908, Γ 1 4 36256, Γ 1 3 26789, Γ 2 5 22182, Γ 1 2 # (cid:26) (cid:29) (cid:31)(cid:2)(cid:0) # (cid:26) (cid:29) (cid:31)(cid:3)(cid:0) # (cid:26) (cid:29) (cid:31)(cid:4)(cid:0) # (cid:26) (cid:29) (cid:31) (cid:19) - 17725, Γ 3 5 14892, Γ 2 3 13541, Γ 3 4 12254, and (cid:21)(cid:5)(cid:0) # (cid:26) (cid:29) (cid:31)(cid:6)(cid:0) # (cid:26) (cid:29) (cid:31)(cid:6)(cid:0) # (cid:26) (cid:29) (cid:31)(cid:6)(cid:0) # Γ 4 5 11642. The Stirling’s formula was found by de Moivre which (cid:26) (cid:29) (cid:31)(cid:7)(cid:0) # approximates the gamma function. The gamma function expansions is (cid:14) (cid:1) (cid:1) (cid:1)(cid:5)(cid:1) (cid:1) (cid:2) (cid:2) 1(cid:2) 2 3(cid:1)(cid:5)(cid:1) (cid:1) (cid:2) (cid:3) Γ 1 lim (cid:18) (cid:18) 2 (cid:26) % (cid:31) (cid:19) 1 2 (cid:26) (cid:31) (cid:1)(cid:9)(cid:8) (cid:28) (cid:26) % (cid:31) (cid:26) % (cid:31) (cid:26) % (cid:18) (cid:31) and the gamma function of negative numbers can be obtained from Γ (cid:25) (cid:21) 3 (cid:26) (cid:25) (cid:15) (cid:31) (cid:19) Γ sin # (cid:26) (cid:31) (cid:15) (cid:26)(cid:10)(cid:15) (cid:31) (cid:21) (cid:15) The incomplete gamma function is (cid:14) (cid:2) (cid:3) (cid:3) Γ (cid:27) 1d (cid:27) (cid:14) (cid:28) 1d 4 (cid:26)(cid:10)(cid:15) (cid:31) (cid:19) (cid:8) / (cid:31) ! (cid:8) (cid:31) (cid:19) (cid:8) / (cid:31) ! (cid:8) (cid:31) (cid:26) (cid:31) 0 (cid:30) (cid:30) and the normalised or regularised incomplete gamma function is (cid:2) (cid:3) Γ (cid:26) (cid:15) (cid:31) Γ (cid:26)(cid:10)(cid:15) (cid:31) Statistics (cid:2) (cid:14) (cid:2) (cid:3) binomPoiaislsdonistdriibsutrtiibount,ion(cid:2),(cid:3)de(cid:3) (cid:2)ned bCy(cid:14) (cid:10) (cid:14)(cid:26) 1(cid:12)(cid:11) (cid:31) (cid:19)(cid:14)(cid:13)(cid:15)(cid:11)(cid:14) I (cid:8)(cid:17)(cid:16),(cid:29) w!h(cid:18) eIn(cid:19)0(cid:20)(cid:28) (cid:18),goisesthtoe in(cid:2)nity, goes to zero(cid:10),,(cid:26) w(cid:0)h(cid:21)i(cid:10)(cid:22)le(cid:31) (cid:19)(cid:23)(cid:22) (cid:25)(cid:24) .(cid:26) H(cid:25) e(cid:24)r(cid:31)(cid:26)e(cid:22) (cid:8) is(cid:19)0(cid:30) (cid:20)(cid:22)(cid:25)th(cid:27) e proba(cid:0) bility of (cid:24) (cid:0) (cid:24) (cid:19)(cid:28)(cid:11) (cid:24) success of each trial. It is used when counting the number of occur› (cid:2) (cid:14) (cid:2) rences of a random event. Analogousl(cid:3)y Poisson point process, which ha(cid:2)s (cid:14) (cid:14) (cid:14) ! I , is the binomial point process, (cid:10),(cid:26) (cid:19) (cid:0) (cid:26)(cid:21)(cid:29) (cid:31) (cid:31))(cid:19) (cid:13)(cid:11),((cid:29)(cid:22)( (cid:8)(cid:17)(cid:16)(cid:31)(cid:30) (cid:25)(cid:30) (cid:29) (cid:18) (cid:19)0(cid:20)(cid:28) (cid:18) C 1 I , when the volume goes to in(cid:2)nity, (cid:10)w(cid:26)hil(cid:19)e(cid:0) (cid:26)!(cid:29) (cid:31) (cid:31) (cid:19) (cid:22) . H(cid:24) e(cid:26)re(cid:25) (cid:30) (cid:24) (cid:31) (cid:22) (cid:8) (cid:19)0(cid:20)(cid:22)"(cid:27) is the probability o(cid:2)f points within (cid:0) (cid:29) ((cid:2) ( (cid:19)#(cid:11) (cid:24) (cid:19) ((cid:29)(cid:22)((cid:29) ((cid:2) ( (cid:2) being placed in , and the density or intensity of points. (cid:29)%$ (cid:2)&$(cid:23)’ (cid:4) (cid:11) Therefore the density of point of a Poisson point process is constant by de(cid:2)nition. A point process is a procedure which generates points on a domain within a space of dimensions. (cid:1) The Poisson point process thus derived has the properties that 0 1 for 0 , lim 1 0, mu› tua(cid:5)(cid:23)lly(cid:10) (cid:22) i(cid:15)(cid:15)n (cid:18)d+ e0p(cid:5)endent and(cid:5) ((cid:29)(cid:22)( (cid:5)&( (cid:30) (cid:25)(cid:30)(cid:8) 0(cid:10),(cid:26) (cid:0)w(cid:26)!h(cid:29) e(cid:31)(cid:3)n) (cid:31)ar(cid:19)e dis(cid:0)jo(cid:26)(cid:21)i(cid:29)(cid:23)n(cid:3)t(cid:31), and (cid:0) (cid:26) (cid:14) (cid:29) (cid:3) (cid:31))(cid:19)+* (cid:0) (cid:26)!(cid:29) (cid:3) (cid:31) (cid:6) (cid:3) lim 0 1 (cid:22) 1 (cid:1) 1. (cid:22) (cid:8) ,(cid:10) (cid:26) (cid:0) (cid:26)(cid:21)(cid:29) (cid:31)-) (cid:31) (cid:29).(cid:10) (cid:26) (cid:0) (cid:26)(cid:21)(cid:29) (cid:31) (cid:19) (cid:31) (cid:19) (cid:30) (cid:25)(cid:30) VaenSryayudhya,Editor January2005 13 TyabandhaJournalofArtsandScience Vol. 2,No. 1 (cid:2) The weighted mean of a gro(cid:2) up of data is x and the weighted var(cid:2)iance is 2 x 2 , where(cid:19) * is(cid:3)(cid:1)(cid:0)th(cid:3) e(cid:3) (cid:29)o(cid:0)ccurrence frequency of and(cid:2) (cid:2) (cid:19) * .(cid:3)(cid:3)(cid:0)L(cid:3)i(cid:26)ke(cid:3) w(cid:25) ise(cid:31) t(cid:29)h(cid:0)e th›mom(cid:0) (cid:3)ent around the av› erageis (cid:9)(cid:5)(cid:4) (cid:2) (cid:19) *(cid:3) (cid:3)(cid:6)(cid:0) (cid:3) (cid:26) * (cid:25) (cid:3)(cid:1)x(cid:0) (cid:31)(cid:3)(cid:4) (cid:19)(cid:29) (cid:0) (cid:0), while the (cid:3) th›m(cid:3)oment aroundthe origin is m(cid:9)(cid:5)(cid:7)(cid:4)om(cid:19) e*nt(cid:3)s(cid:0) a(cid:3) re(cid:4) (cid:29) (cid:0) (cid:26) cf S0p,iegel, 1975(cid:31) . Som2,e relations am3ong thes2e va2,riaonuds (cid:9) 1 (cid:19) (cid:9) 2 (cid:19) (cid:9) 2(cid:7) (cid:25) (cid:9) 1(cid:7) (cid:9) 3 (cid:19) (cid:9) 3(cid:7) (cid:25) (cid:9) 1(cid:7) (cid:9) 2(cid:7) % (cid:9) 1(cid:7) 4 6 2 3 4. (cid:9) 4 (cid:19) (cid:9)(cid:5)4(cid:7) (cid:25) (cid:9)(cid:8)1(cid:7) (cid:9)(cid:5)3(cid:7) % (cid:9)(cid:5)1(cid:7) (cid:9)(cid:5)2(cid:7) (cid:25) (cid:9)(cid:5)1(cid:7) The variance when normalised by 1 gives the best unbiased (cid:0) (cid:25) estimated variance if the sample has a normal distribution. On the other hand the variance which is normalised by is identical with the (cid:0) second moment of the sample about its mean. Phase transition In the Ising model each spin has two possible states, that is up and down, and the Hamiltonian is where the summa› 0 tion is over the nearest neighbo(cid:9)urs(cid:19)(cid:11). (cid:10)Sin*(cid:13)ce(cid:12) (cid:3)i(cid:20)(cid:11)(cid:15)t(cid:14) h(cid:2)a(cid:3)(cid:16)s(cid:2)(cid:22)(cid:11)been exactly solved, the Ising model provides a good model for the understanding of phase transition. Thismodelcanrepresentthetransitionfromferro›toparam› agnetic atthe critical temperaturewherethe correlation length becomes in(cid:2)nite. Characteristic to the Ising model is the peakin the speci(cid:2)c heat at the critical temperature. The two›dimensional xy model is a model of spins con(cid:2)ned to a plane, the Hamiltonian of which is cos . This model can represent the superconduct(cid:9)ing(cid:19)(cid:17)an(cid:10) 0d*(cid:18)th(cid:12)e(cid:3) (cid:20)(cid:11)(cid:15)s(cid:14)uper(cid:3)(cid:26) (cid:24)u(cid:3)id(cid:25) (cid:2)(cid:24) (cid:11)(cid:30)lm(cid:31) s. For this model there is no phase transition showing long›range ordering. One example is the two›dimensional Coulomb gas model where the vortex›antivortex pairs, which are bound to each other at low temper› ature, increases in number as the temperature increases and become separated at the KT temperature that marks the phase transition. It had been generally believed that no phase transition can ex› ist for the xy model when Kosterlitz et al 1973 showed that there is (cid:26) (cid:31) another kind of phase transition, arisen from the topological excita› tion of vortex›antivortex pairs instead of from the long›range order› ing found in a spontaneous magnetisati(cid:27)on. They consider the two› dimensional model of gas with charges (cid:2) where the interaction po› tential is r r is 2(cid:2) (cid:2) ln r r r 2 when , and 0 (cid:19) (cid:26) ( (cid:3) (cid:25) (cid:11)(cid:2) ((cid:31) (cid:25) (cid:3)(cid:14) (cid:11) ((cid:17)(cid:26) (cid:3) (cid:25) (cid:11)(cid:30)(cid:31) (cid:29) 0( % (cid:20) (cid:3) # (cid:3) 0 when . The problem is reduced to that of solving an equation (cid:3)(cid:6)(cid:5) (cid:3) 0 of the form d d (cid:22)(cid:21) . The application mentioned there is in (cid:26) (cid:12) (cid:29) (cid:31) (cid:19) (cid:25) (cid:8) the xy model of magnetism, the solid›liquid transition, and the neutral (cid:30) super(cid:3)uid, but not in a superconductor and a Heisenberg ferromagnet. The frustrated xy model, the Hamiltonian of which is cos (cid:9) (cid:19)(cid:13)(cid:10) 0 (cid:23) (cid:26) (cid:24) (cid:3) (cid:25) (cid:24) (cid:11) (cid:25) (cid:6)(cid:2)(cid:3)(cid:11)(cid:7)(cid:31) (cid:12) (cid:3) (cid:20)(cid:11)(cid:15)(cid:14) 14 January2005 VaenSryayudhya,Editor