ITI CENTRAL SCHOOL

MATH PROJECT

Topic

MODEL MAKING AND

APPLICATIONS OF

POLYHEDRON

POLYHEDRONINTRODUCTION TETRAHEDRON DODECAHEDRON HEXAHEDRON

OCTAHEDRON

ICOSAHEDRON

GREAT DODECAHEDRONPYRAMID

PRISM

PENTAGONAL BIPYRAMIDTRIAUGMENTED TRIANGULAR PRISM

VIDEO

POLYHEDRONA polyhedron (plural

polyhedra or polyhedrons) is often defined as a geometric solid with flat faces and straight edges.

The word polyhedron comes from the Classical Greek from poly from πολύεδρον, -, stem of πολύς, meaning "many," + -edron, from of εδρον, meaning "base", "seat", or "face“.

Polyhedron

In geometry, a polyhedron is traditionally a three-dimensional shape that is made up of a finite number of polygonal faces which are parts of planes; the faces meet in pairs along edges which are straight-line segments, and the edges meet in points called vertices. Cubes, prisms and pyramids are examples of polyhedra.

The polyhedron surrounds a bounded volume in three-dimensional space; sometimes this interior volume is considered to be part of the polyhedron, sometimes only the surface is considered, and occasionally only the skeleton of edges.

Naming of Polyhedron

• Polyhedra are often named according to the number of faces. The naming system is again based on Classical Greek. Eg: tetrahedron (4), pentahedron (5), hexahedron (6).

• Often this is qualified by a description of the kinds of faces present. Eg: Rhombic dodecahedron, Pentagonal dodecahedron.

• Other common names indicate that some operation has been performed on a simpler polyhedron. Eg: Truncated cube looks like a cube with its corners cut off, and has 14 faces.

• Some special polyhedra have grown their own names over the years, such as Miller's monster or the Szilassi polyhedron.

Characteristics of PolyhedronEdges have two important characteristics (unless

the polyhedron is complex):i. An edge joins just two vertices. ii. An edge joins just two faces. iii. These two characteristics are dual to each other.

Euler characteristici. The Euler characteristic χ relates the number of

vertices V, edges E, and faces F of a polyhedron:χ = V - E + F.

ii. For a simply connected polyhedron, χ = 2. For a detailed discussion, see Proofs and Refutations by Imre Lakatos.

Dualityi. For every polyhedron there is a dual polyhedron

having faces in place of the original's vertices and vice versa. For a convex polyhedron the dual can be obtained by the process of polar reciprocation.

Types of polyhedron Traditional polyhedra

i. Symmetrical polyhedra

ii. Uniform polyhedra iii. Noble polyhedra

Deltahedra Johnson solids Other important

families of polyhedrai. Pyramidsii. Stellations and

facettingsiii. Zonohedraiv. Toroidal polyhedrav. Orthogonal

polyhedra

vi. Apeirohedravii. Complex polyhedraviii. Spherical polyhedraix. Curved polyhedra

Non-geometric polyhedrai. Topological polyhedraii. Abstract polyhedraiii. Graphical polyhedra

Generalisation of Polyhedra

i. Tessellations or tilingsii. Hollow faced or

skeletal polyhedraiii. Curved spacefilling

polyhedra

Tetrahedron

• In geometry, a tetrahedron (plural: tetrahedra) is a polyhedron composed of four triangular faces, three of which meet at each vertex.

• A regular tetrahedron is one in which the four triangles are regular, or "equilateral", and is one of the Platonic solids. The tetrahedron is the only convex polyhedron that has four faces.

• The tetrahedron is one kind of pyramid, which is a polyhedron with a flat polygon base and triangular faces connecting the base to a common point.

• In the case of a tetrahedron the base is a triangle (any of the four faces can be considered the base), so a tetrahedron is also known as a triangular pyramid.

• For a regular tetrahedron of edge length a:

Base plane area:

Surface area:

Height:

Volume:

FORMULAS FOR REGULAR TETRAHEDRON

Verification of Euler’s Rule

• Number of Faces (F): 4• Number of Vertices (V): 4• Number of edges (E): 6• According to Euler’s rule : E+2 = V+F

Substituting the values: LHS : E+2= 6+2=8

RHS: V+F = 4+4 = 8

Therefore LHS=RHS

Geometric relations

• A tetrahedron is a 3-simplex. Unlike the case of the other Platonic solids, all the vertices of a regular tetrahedron are equidistant from each other (they are the only possible arrangement of four equidistant points in 3-dimensional space).

• A truncation process applied to the tetrahedron produces a series of uniform polyhedra.

• Truncating edges down to points produces the octahedron as a rectified tetahedron.

• The process completes as a birectification, reducing the original faces down to points, and producing the self-dual tetrahedron once again.

The volume of this tetrahedron is 1/3 the volume of the cube.

Combining both tetrahedra gives a regular polyhedral compound called the Stella octangula, whose interior is an octahedron.

A tetrahedron is a triangular pyramid, and the regular tetrahedron is self-dual.

• Water is an example as well, since the oxygen atom is not only surrounded by two hydrogen atoms, but by two lone pairs as well.

• The symmetry isn't perfect in this case because of the in equivalency of the surrounding groups.

• The angle of HOH is 105 degrees.

Water Molecule

• One of the leading journals in organic chemistry is called Tetrahedron. The ammonium ion is another example.

• In an ammonium ion, the nitrogen atom forms four covalent bonds (including one coordinate covalent bond), instead of three as in ammonia, forming a structure of tetrahedron.

Ammonium molecule

Applications

The tetrahedron shape is seen in nature in covalent bonds of molecules. All sp3-hybridized atoms are surrounded by atoms lying in each corner of a tetrahedron.

• In a methane molecule (CH4) the four hydrogen atoms surround the carbon atom with tetrahedral symmetry.

• Methane's bond angles are 109.5 degrees.

Methane Molecule

ApplicationsElectronics

When the resistors are placed in the form of a tetrahedron, the resistance would be reduced by a one ohm resistor.

The resistance between any two vertices would be 1/2 ohm.

Tetrahedral resistors

Computational fluid dynamics (CFD) is one of the branches of fluid mechanics that uses numerical methods and algorithms to solve and analyze problems that involve fluid flows.

Complex shapes are often broken down into a mesh of irregular tetrahedra in preparation for finite element analysis and computational fluid dynamics studies.

Applications

The Pyramorphix is a tetrahedral puzzle similar to the Rubik's Cube.

the puzzle is a specially shaped 2×2×2 cube.

Four of the cube's corners are reshaped into pyramids and the other four are reshaped into triangles.

Applications

The Pyraminx is a tetrahedral puzzle similar to the Rubik's Cube.

Pyraminx is  divided into 4 axial pieces, 6 edge pieces, and 4 trivial tips.

It can be twisted along its cuts to permute its pieces.

• Pyraminx

Games Especially in role-

playing, this solid is known as a 4-sided die, one of the more common polyhedral dice, with the number rolled appearing around the bottom or on the top vertex.

4 sided die

Color TranslationUsed in color space

conversion algorithms specifically for cases in which the luminance axis diagonally segments the color space (e.g. RGB, CMY).

Hexahedron

• In geometry, a Hexahedron or cube is a three-dimensional solid object bounded by six square faces, facets or sides, with three meeting at each vertex. The cube can also be called a regular hexahedron and is one of the five Platonic solids. It is a special kind of square prism, of rectangular parallelepiped and of trigonal trapezohedron. The cube is dual to the octahedron. It has cubical symmetry (also called octahedral symmetry).

• A cube is the three-dimensional case of the more general concept of a hypercube.

• It has 11 nets. If one were to colour the cube so that no two adjacent faces had the same colour, one would need 3 colours.

• If the original cube has edge length 1, its dual octahedron has edge length .

Formulas associated with Cube

• For a cube centered at the origin, with edges parallel to the axes and with an edge length of 2, the Cartesian coordinates of the vertices are: (±1, ±1, ±1) .

• For a cube of edge length a,• surface area :6a2 • Volume: a3 • face diagonal : root 2a• space diagonal : root 3a• Faces=6, vertices=8, edges=12

Geometric relations• As the volume of a cube is the third power of its sides a×a×a,

third powers are called cubes, by analogy with squares and second powers.

• A cube has the largest volume among cuboids (rectangular boxes) with a given surface area. Also, a cube has the largest volume among cuboids with the same total linear size (length + width + height).

• The cube is unique among the Platonic solids for being able to tile Euclidean space regularly. It is also unique among the Platonic solids in having faces with an even number of sides and, consequently, it is the only member of that group that is a zonohedron (every face has point symmetry).

• The cube can be cut into 6 identical square pyramids. If these square pyramids are then attached to the faces of a second cube, a rhombic dodecahedron is obtained.

Applicati on• James Clar’s cool 3D cubes which could display images using a

grid of about 1000 LEDs connected to a computerized controller. • While Clar’s cubes are simply amazing, they can only display

monochrome images. • Now, a company in China has taken the same concept and

created a multi-color version.Created by Chinese display manufacturer Seekway, this prototype 3D LED display cube was created using a 16×16x16 grid of interconnected color LEDs (that’s a whopping 4096 individual diodes). The system is capable of displaying animations at up to 30 frames per second and each dot can be individually addressed for both color and intensity.

• Now that the prototype is complete, Seekway is gearing up to produce versions with grids as large as 48×48x48, which will require more than 110,000 LEDs.

• The rigid hexahedral coordination is realized only in the CsCl crystal structure. Naturally, in the crystal the atoms B as well as the atoms A appear as center atoms with respect to the surrounding atoms. In the crystal, the number of atoms A is equal to the number of atoms B.

• In the realization as an isolated molecule, the hexahedron (cube) twists to form the square antiprism. The center atom A is connected to 8 complexes B at the corners of the twisted structure. This results in less strain on the ligands. Even though each corner lies on the same sphere as in the in case of the untwisted hexahedron, the distance between the repulsive complexes B is increased and thus favoured. Still, the coordination number 8 is not very common. A typical example is the complex [Mo(CN)8]4 -.

Octahedron• In geometry, an

octahedron (plural: octahedra) is a polyhedron with eight faces. A regular octahedron is a Platonic solid composed of eight equilateral triangles, four of which meet at each vertex.

• It is a 3-dimensional cross polytope.

• The octahedron represents the central intersection of two tetrahedra

• The octahedron is unique among the Platonic solids in having an even number of faces meeting at each vertex. Consequently, it is the only member of that group to possess mirror planes that do not pass through any of the faces.

• Using the standard nomenclature for Johnson solids, an octahedron would be called a square bipyramid.

Formulas related to Octahedron

• The area A and the volume V of a regular octahedron of edge length a are:

• Radius of a circumscribed sphere:• The radius of an inscribed sphere:• The midradius, which touches • the middle of each edge :• An octahedron can be placed with its center

at the origin and its vertices on the coordinate axes; the Cartesian coordinates of the vertices are then ( ±1, 0, 0 ); ( 0, ±1, 0 ); ( 0, 0, ±1 ).

Verification of Euler’s Rule

Number of Faces (F): 8 Number of Vertices (V): 6 Number of Edges (E): 12 According to Euler’s formula :

E+2=F+VSubstituting the values: LHS: E+2

=12+2=14 RHS: F+V

= 8+6= 14Therefore: LHS=RHS

Geometric Relations

• The vertices of the octahedron lie at the midpoints of the edges of the tetrahedron, and in this sense it relates to the tetrahedron in the same way that the cuboctahedron and icosidodecahedron relate to the other Platonic solids.

• Octahedra and tetrahedra can be alternated to form a vertex, edge, and face-uniform tessellation of space, called the octet truss by Buckminster Fuller.

• This is the only such tiling save the regular tessellation of cubes, and is one of the 28 convex uniform honeycombs. Another is a tessellation of octahedra and cuboctahedra.

Cuboctahedron

• One can also divide the edges of an octahedron in the ratio of the golden mean to define the vertices of an icosahedron.

• This is done by first placing vectors along the octahedron's edges such that each face is bounded by a cycle, then similarly partitioning each edge into the golden mean along the direction of its vector.

• There are five octahedra that define any given icosahedron in this fashion, and together they define a regular compound.

Tetrahemihexahedron

APPLICATIONS• Especially in roleplaying

games, this solid is known as a "d8", one of the more common non-cubical dice.

• Many puzzle games like Jaap’s puzzle introduce octahedral tetraminx also called as Christophe's jewel.

• This is a regular octahedron, of which each triangular face is divided into 9 identical smaller triangles, three to a side.

• A move consist of rotating a vertex; either just a tip (one triangle to a side) or a larger part (two triangles to a side).

TETRAMINX

• A Rubik's Snakeis a toy with twenty-four wedges identically shaped liked prisms, specifically right isosceles triangular prisms. The wedges are connected, by spring bolts, such that they can be twisted, but not separated. Through this twisting, the Rubik's Snake can attain octahedral positions.

Rubik’s Snakes

• Natural crystals of diamond, alum or fluorite are commonly octahedral.

• Diamonds crystallize as octhedrons, and commonly exhibit habits that contain two or more of these forms.  Although we find diamonds at the Earth’s surface, they form deep within the Earth.

Fluorite Octahedron:

Octahedra in music

• Six musical notes can be arranged on the vertices of an octahedron in such a way that each edge represents a consonant dyad and each face represents a consonant triad.

• The following polyhedra are combinatorially equivalent to the regular polyhedron. They all have six vertices, eight triangular faces, and twelve edges that correspond one-for-one with the features of a regular octahedron.

Alum Octahedron and Cluster, both grown using the supercooling method. These crystals of Alum are cubic in structure, and grown into the ideal form of an octahedron. Diamond is of the cubic form and also grows into the octahedron.

Before After

Icosahedron

• In geometry, an icosahedron (Greek: εικοσάεδρον, from eikosi twenty + hedron seat; pronounced /ˌaɪkɵsəˈhiːdrən/ or /aɪˌkɒsəˈhiːdrən/; plural: -drons, -dra /-drə/) is a regular polyhedron with 20 identical equilateral triangular faces, 30 edges and 12 vertices. It is one of five Platonic solids.

• The following Cartesian coordinates define the vertices of an icosahedron with edge-length 2, centered at the origin:(0, ±1, ±φ) ,(±1, ±φ, 0) ,(±φ, 0, ±1) , where φ = (1+√5)/2 is the golden ratio (also written τ). Note that these vertices form five sets of three concentric, mutually orthogonal golden rectangles, whose edges form Borromean rings.

• The surface area A and the volume V of a regular icosahedron of edge length a are:

• The radius of a circumscribed sphere:• The radius of an inscribed sphere : • The midradius, which

touches the middle of each edge: where (also called τ) is the golden ratio.

Formula Related to Icosahedrons

Geometric Relations and Facts

• Icosahedron has 43,380 nets. • If one were to color the icosahedron such that no two

adjacent faces had the same color, one would need to use 3 colors.

• When an icosahedron is inscribed in a sphere, it occupies less of the sphere's volume (60.54%) than a dodecahedron inscribed in the same sphere (66.49%).

• The icosahedron is unique among the Platonic solids in possessing a dihedral angle not less than 120°. Its dihedral angle is approximately 138.19°. Thus, just as hexagons have angles not less than 120° and cannot be used as the faces of a convex regular polyhedron because such a construction would not meet the requirement that at least three faces meet at a vertex and leave a positive defect for folding in three dimensions, icosahedra cannot be used as the cells of a convex regular polychoron because, similarly, at least three cells must meet at an edge and leave a positive defect for folding in four dimensions.

• An icosahedron can also be called a gyroelongated pentagonal bipyramid. It can be decomposed into a gyroelongated pentagonal pyramid and a pentagonal pyramid or into a pentagonal antiprism and two equal pentagonal pyramids.

Applications

• Many viruses, e.g. herpes virus, Cirgona virus, have the shape of an icosahedron. Viral structures are built of repeated identical protein subunits and the icosahedron is the easiest shape to assemble using these subunits. A regular polyhedron is used because it can be built from a single basic unit protein used over and over again; this saves space in the viral genome.

Cirgonia virus

Electron micrograph of Herpes simplex virus

• In 1904, Ernst Haeckel described a number of species of Radiolaria, which resemble the shape of icosahedra, whose skeleton is shaped like a regular icosahedron.

• It produces an intricate mineral skeletons, typically with a central capsule dividing the cell.

Radiolara Protozoa

Benzene:• Benzene, or benzol,

is an organic chemical compound with the molecular formula C6H6.

• It is sometimes abbreviated Ph–H.

• Benzene is a colorless and highly flammable liquid with a sweet smell and a relatively high melting point.

• The closo-carboranes are chemical compounds with shape very close to isosahedron. Icosahedral twinning also occurs in crystals, especially nanoparticles.

Conversion of Benzene to fullerene

Icosahedrons in games• In several roleplaying games,

such as Dungeons & Dragons, the twenty-sided die (d20 for short) is commonly used in determining success or failure of an action. This die is in the form of a regular icosahedron. It may be numbered from "0" to "9" twice (in which form it usually serves as a ten-sided die, or d10), but most modern versions are labeled from "1" to "20". See d20 System.

• An icosahedron is the three-dimensional game board for Icosagame, formerly known as the Ico Crystal Game.

• An icosahedron is used in the board game Scattergories to choose a letter of the alphabet. Six little-used letters, such as X, Q, and Z, are omitted.

• Inside a Magic 8-Ball, various answers to yes-no questions are printed on a regular icosahedron.

• Icosahedral structures play an important role in the chemistry of boron, e.g. by a reaction of boroydride and diborane:2[BH4] - + 5B2H6 ---> [B12H12] - + 13H2

• The boron atoms at the corners of the icosahedron are each bound to 5 adjacent boron atoms and to one external hydrogen (not shown in the model) by a terminal B-H bond. The twofold negative charge of the complex may attract metal ions, e.g two K+ ions. This icosahedron may by itself form the center unit for a icosahedron of 12 surrounding identical icosahedrons.

Great Dodecahedron

• In geometry, the great dodecahedron is a Kepler-Poinsot polyhedron, with Schläfli symbol {5,5/2}. It is one of four nonconvex regular polyhedra. It is composed of 12 pentagonal faces (six pairs of parallel pentagons), with five pentagons meeting at each vertex, intersecting each other making a pentagrammic path.

Application• Alexander's Star is

a puzzle similar to the Rubik's Cube, in the shape of a great dodecahedron.

• Alexander's Star was invented by Adam Alexander, an American mathematician, in 1982. It was patented on 26 March 1985, with US patent number 4,506,891, and sold by the Ideal Toy Corporation.

• There are 30 edges, each of which can be flipped into two positions.

PYRAMID• In geometry, a pyramid is a

polyhedron formed by connecting a polygonal base and a point, called the apex. Each base edge and apex form a triangle. It is a conic solid with polygonal base.

• The regular tetrahedron, one of the Platonic solids, is a triangular pyramid all of whose faces are equilateral triangles. Besides the triangular pyramid, only the square and pentagonal pyramids can be composed of regular convex polygons, in which case they are Johnson solids.

The Square based Pyramid

Formulas and Net• The Johnson square

pyramid can be characterized by a single edge-length parameter a. The height H (from the midpoint of the square to the apex), the surface area A (including all five faces), and the volume V of such a pyramid are:

The net of the square based pyramid:

• In geometry, a triangular prism or three-sided prism is a type of prism; it is a polyhedron made of a triangular base, a translated copy, and 3 faces joining corresponding sides.

• A right triangular prism is semiregular if the base faces are equilateral triangles, and the other three faces are squares.

Prism

FormulasVOLUME• The volume of a prism is the product of the area of the

base and the distance between the two base faces, or the height (in the case of a non-right prism, note that this means the perpendicular distance).

• The volume is therefore: V=B.hSurface area• The surface area of a right prism is , where B is the area

of the base, h the height, and P the base perimeter.• The surface area of a right prism whose base is a

regular n-sided polygon with side length S and height h is therefore:

• In optics, a prism is a transparent optical element with flat, polished surfaces that refract light. The exact angles between the surfaces depend on the application.

• A prism can be used to break light up into its constituent spectral colors (the colors of the rainbow). Prisms can also be used to reflect light, or to split light into components with different polarizations.

• The moment of inertia of triangular prism (equilateral triangles with side 2a, parallel to xy-plane), mass M. It's centered at the origin with long side parallel to z-axis.  and moment of inertia about the z-axis, and without doing integrals explain two products of inertia (Ixy, Ixz, I suppose)2. Relevant equationsThe integral equation for Izz=int[(M/V)*(x^2+y^2)dV]where V=volume, or in cylindrical cords. It can be written as (x^2+y^2)=r^2, where r=distance from z-axis

Pentagonal Bipyramid• In geometry, the pentagonal bipyramid (or

dipyramid) is third of the infinite set of face-transitive bipyramids.

• The set of bipyramids is the dual of the uniform prisms.• It is also a Johnson solid (J13), constructed of regular

polygons. It can be seen as two pentagonal pyramids (J2) connected along their bases.

• As a Johnson solid, it is a convex deltahedron. Although it is face-transitive, it is not a Platonic solid because some vertices have four faces meeting and others have five faces. The Johnson solid 13 is made of 10 equilateral triangles.

Application

• In chemistry a pentagonal bipyramid (or dipyramid) is a molecular geometry with one atom at the centre with seven ligands at the corners of a pentagonal dipyramid. A perfect pentagonal bipyramid belongs to the molecular point group D5h . The pentagonal bipyramid is a case where bond angles surrounding an atom are not identical. Other seven coordinate geometries include the mono-capped octahedron and mono-capped trigonal prism. A variety of transition metal complexes adopt heptacoordination, but the symmetry is usually lower than D5h.

• Iodine heptafluoride, also known as iodine(VII) fluoride or even iodine fluoride, is the compound IF7.It has an unusual pentagonal bipyramidal structure, as predicted by VSEPR theory.

triaugmented triangular prism

• In geometry, the triaugmented triangular prism is one of the Johnson solids (J51). As the name suggests, it can be constructed by augmenting a triangular prism by attaching square pyramids (J1) to each of its three equatorial faces. It is a deltahedron.

• The dual of the triaugmented triangular prism is an order-5 associahedron. This transparent image shows its three square, and six congruent irregular pentagonal faces. Edges are colored to distinguish the 3 different edge lengths.

• In chemistry, a coordination complex or metal complex, is a structure consisting of a central atom or ion (usually metallic), bonded to a surrounding array of molecules or anions (ligands, complexing agents). For example: Cisplatin, PtCl2(NH3)2A platinum atom with four ligands

• When two ligands are mutually adjacent they are said to be cis, when opposite to each other, trans, three identical ligands occupy one face of an Triaugmented triangular prism.

• For example : trans-[CoCl2(NH3)4]+

• The use of ligands of different types which results in irregular bond lengths; the coordination atoms do not follow a points-on-a-sphere pattern.

• For example : Tri-capped trigonal prismatic (Triaugmented triangular prism) for nine coordination of Λ-[Fe(ox)3]3−

A Chinese rings puzzle, a recursive puzzle related to the Towers of Hanoi problem: the typical 9-ring version in his collection requires 341 moves

to solve.

This beautiful polyhedron was built by Flickr user fdecomite using George Hart's slide-together technique for making complex 3D paper shapes.

Icosahedron: origami shape

Origami tetrahedron, This is the combination of many tetrahedron’s skeletal edges

Origami icosahedrons