Pure mathematics is, in its way, the poetry of logical ideas. --- Albert Einstein ...Economics is nothing but Mathematics -Dr.Ahsan Abbass...Symmetry is Ornament of Mathematics-Zulfiqar Ali Mir...Law of Nature are But Mathematical Thoughts of God - Euclid (Father of Geometry)...Mathematics is about Number and pattern among these nos.-Sir Zulfiqar A Mir,... Number Theory is Foundation of Mathematics-Sir Zulfiqar Ali Mir

Wednesday, 28 November 2012

The Failure of Mathematics?

The Failure of Mathematics?

Mathematics has come in for criticism recently, that somehow it was the root cause of the credit crunch and the collapse of the world economy. It was the misuse of mathematics, by bosses who didn’t understand it, that got us into the mire.

Actually, bosses do possess the tools; they simply refuse to use them. And then we wonder when it goes wrong.

The refusal to take mathematics seriously is the cause of all sorts of errors.

  • We assume things are static when they are dynamic.
  • We assume relationships between variables to be linear (and predictable) when they are not.
  • We ignore ranges of possibility and use single-point average values instead—the flaw of averages.
  • We assume that different uncertainties are independent, and get caught by surprise when we discover they are not.
  • We confuse cause and symptom and treat the latter, and are surprised when the former does not go away.
  • We assume that absence of evidence is evidence of absence.
  • We let vanishingly small risks bias our perception of the world (and let unwarranted fear guide our behavior), whilst ignoring the everyday risks that kill lots of us all the time.
The mathematics is sound. It’s the application that’s at fault. The failure of mathematicians—but not, —is to let get away with it.

Friday, 2 November 2012

Worksheets IGCSE and O Level Mathematics

Worksheets 
IGCSE 
Mathematics

Note: Worksheet & Resource Pack is available on Separate topic
Price per topic = US $1
Send you mail order on : 
Accountmatics@ymail.com



SHEETS 
  • Extended

Trigonometry, Circle, Bearing
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Locus, Coordinate Geometry, Vector
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Locus and Coordinate Geometry
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  • Core
Statistics with Tally

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Mensuration with Unit Cost

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Sequences

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Algebra - Factorization, Subject, Indices

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Transformation

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Equation

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Perimeter

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Bearing

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Locus

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PAST PAPERS WITH MARK SCHEME
(2005-2012)

IGCSE
Click 

Thursday, 18 October 2012

Wednesday, 12 September 2012

IGCSE PAST PAPERS (Maths)

IGCSE 
PAST PAPERS  
MATHEMATICS
580
TOPICAL

PROBABILITY


IN-EQUALITY


VEN DIAGRAM

Read

JUNE 2006

READ

Sunday, 9 September 2012

Simplex Method


Simplex algorithm


In mathematical optimizationDantzig's simplex algorithm (or simplex method) is a popular algorithm for linear programming.The journal Computing in Science and Engineering listed it as one of the top 10 algorithms of the twentieth century .


Example

Consider the linear program
Minimize
Z = -2 x - 3 y - 4 z\,
Subject to
\begin{align}
 3 x + 2 y + z &= 10\\
 2 x + 5 y + 3 z &= 15\\
 x,\, y,\, z &\ge 0
\end{align}
This is represented by the (non-canonical) tableau

  \begin{bmatrix}
    1 & 2 & 3 & 4 &  0 \\   
    0 & 3 & 2 & 1 & 10 \\
    0 & 2 & 5 & 3 & 15
  \end{bmatrix}
Introduce artificial variables u and v and objective function W = u + v, giving a new tableau

  \begin{bmatrix}
    1 & 0 & 0 & 0 & 0 & -1 & -1 &  0 \\  
    0 & 1 & 2 & 3 & 4 &  0 &  0 &  0 \\   
    0 & 0 & 3 & 2 & 1 &  1 &  0 & 10 \\
    0 & 0 & 2 & 5 & 3 &  0 &  1 & 15
  \end{bmatrix}
Note that the equation defining the original objective function is retained in anticipation of Phase II. After pricing out this becomes

  \begin{bmatrix}
    1 & 0 & 5 & 7 & 4 & 0 & 0 & 25 \\  
    0 & 1 & 2 & 3 & 4 & 0 & 0 &  0 \\   
    0 & 0 & 3 & 2 & 1 & 1 & 0 & 10 \\
    0 & 0 & 2 & 5 & 3 & 0 & 1 & 15
  \end{bmatrix}
Select column 5 as a pivot column, so the pivot row must be row 4, and the updated tableau is

  \begin{bmatrix}
    1 & 0 &  \tfrac{7}{3} &   \tfrac{1}{3} & 0 & 0 & -\tfrac{4}{3} &   5 \\  
    0 & 1 & -\tfrac{2}{3} & -\tfrac{11}{3} & 0 & 0 & -\tfrac{4}{3} & -20 \\   
    0 & 0 &  \tfrac{7}{3} &   \tfrac{1}{3} & 0 & 1 & -\tfrac{1}{3} &   5 \\
    0 & 0 &  \tfrac{2}{3} &   \tfrac{5}{3} & 1 & 0 &  \tfrac{1}{3} &   5
  \end{bmatrix}
Now select column 3 as a pivot column, for which row 3 must be the pivot row, to get

  \begin{bmatrix}
    1 & 0 & 0 &              0 & 0 &            -1 &            -1  &               0 \\  
    0 & 1 & 0 & -\tfrac{25}{7} & 0 &  \tfrac{2}{7} & -\tfrac{10}{7} & -\tfrac{130}{7} \\   
    0 & 0 & 1 &   \tfrac{1}{7} & 0 &  \tfrac{3}{7} &  -\tfrac{1}{7} &   \tfrac{15}{7} \\
    0 & 0 & 0 &  \tfrac{11}{7} & 1 & -\tfrac{2}{7} &   \tfrac{3}{7} &   \tfrac{25}{7}
  \end{bmatrix}
The artificial variables are now 0 and they may be dropped giving a canonical tableau equivalent to the original problem:

  \begin{bmatrix}
    1 & 0 & -\tfrac{25}{7} & 0 &  -\tfrac{130}{7} \\   
    0 & 1 &   \tfrac{1}{7} & 0 &    \tfrac{15}{7} \\
    0 & 0 &  \tfrac{11}{7} & 1 &    \tfrac{25}{7} 
  \end{bmatrix}
This is, fortuitously, already optimal and the optimum value for the original linear program is −130/7.


EXAMPLE (Part 1): Simplex method

Resolve using the Simplex Method the following problem:
MaximizeZ = f(x,y) = 3x + 2y
subject to:2x + y ≤ 18
 2x + 3y ≤ 42
 3x + y ≤ 24
 x ≥ 0 , y ≥ 0
Are considerated the following phases:
1. Turning the inequalities into equalities
Introduce a slack variable for each restrictions of the type ≤ to turn them into equalities, giving the following linear equation system:
2x + y + r = 18
2x + 3y + s = 42
3x +y + t = 24
2. Equaling the objective function to zero
- 3x - 2y + Z = 0
3. Writing the initial board simplex
At columns will appear all basic variables of the problem and the slack/surplus variables. At rows you can observe, for each restriction the slack variables with its coefficients of obtained equalities, and the last row with the values resulting of substitute the value of every variable at function objetive, and operate just as was explained in the theory to get the left values from the row:
Board I . 1st iteration
   32000
BaseCbP0P1P2P3P4P5
P301821100
P404223010
P502431001
Z 0-3-2000
4. Halt condition
When at the Z row there aren't negatives values , the optimal solution of the problem has been reached. In such a case, the algorithm has finished. If it were not so, the following steps must be executed.
5. Input-output base condition
  1. A. First, we must know the variable that enters into the base. For it we choose that value's column that in the row the Z is the minor of the presents negatives values. In this case would be the variable x (P1) of coefficient - 3.

     If two or more equal coefficients exist that obey the previous condition (tie case), then we will choose that variable that be basic.

    The column of the variable that goes into the base is named pivot column (In green color).
  2. Once the variable that goes into the base was obtained, we are in condition to deduce what will be the variable that goes out . For it, divide each independent term (P0) among the correspondent element of the pivot column, taking care that the result must be bigger than zero, and the minimum of this values is chosen.

     In our case: 18/2 [=9] , 42/2 [=21] y 24/3 [=8]

     If any less or equal to zero element exist, such divide will not be do, or if every elements that belongs to pivot column are zero we are in the case of unbounded solution, and the problem just would be finished (See theory).

     The term of the pivot column that gives the lowest positive quotient in previous division, the 3, right now than 8 is the lower quotient, indicates the row of the slack variable that goes out the base, t (P5). This row is named the pivot row(In green color).

     If two or more quotients are equals when they are being calculated (tie case), make a choice for a non basic variable (if possible).
  3. At intersection of pivot row with pivot column we have the pivot element, 3.
6. Calculating the coefficients of the new board.
The new coefficients of the pivot row , t (P5), are obtained dividing all of the coefficients from such row among the pivot element, 3, that is the one necessary to turn into 1.
Following, with Gaussian reduction we do zeros the remainders terms of that column, with it we get the new coefficients from the other rows including that belong to the objective function row Z.
Also, it can be done in the following way:
Pivot row:
New pivot row = (Old pivot row) / (Pivot)
Remainders rows:
New row = (Old row) - (Coefficients from old row placed at incoming variable's colum) x (New pivot row)
Lets see an example, once the pivot row has been calculated (x's (P1) row at Board II):
Old row P44223010
 ------
Coefficient222222
 xxxxxx
New pivot row811/3001/3
 ======
New row P42607/301-2/3

Board II . 2nd iteration
   32000
BaseCbP0P1P2P3P4P5
P30201/310-2/3
P402607/301-2/3
P13811/3001/3
Z 240-1001
Can be noticed that we have not attained the halt condition because at Z row, there is one negative, -1. We must do another iteration:
  1. The incoming variable is y (P2), in order to be the variable that corresponds to the column where is the -1 coefficient.
  2. B. To calculate the variable that goes out, we divide the terms from last column among the ones correspondent to the new pivot column: 2 / 1/3 [=6] , 26 / 7/3 [=78/7] and 8 / 1/3 [=24]
    and as the lower positive quotient is 6, we have that the variable that goes out is r (P3).
  3. The pivot element, that we must do it 1, is 1/3.
Working of analogous form that before, we obtain the board:
Board III . 3rd iteration
   32000
BaseCbP0P1P2P3P4P5
P2260130-2
P401200-714
P13610-101
Z 300030-1
As you can see, there is a element with minus sign in Z row, - 1, it means that we have not come still to the optimal solution. It is necessary to repeat the process:
  1. The variable that comes to the base is t (P5) , because is the variable that correspond to the -1 coefficient.
  2. To calculate the variable that goes out, we divide the terms from the last colum among the terms correspondent from new pivot colum: 6/(-2) [=-3] , 12/4 [=3], and 6/1 [=6]
    and like the lower positive quotient is 3, we get s (P4) as the variable that goes out from base.
  3. The pivot element, that we must do it 1, is 4.
We get the board:
Board IV . 4th iteration
   32000
BaseCbP0P1P2P3P4P5
P221201-1/21/20
P50300-7/41/41
P133103/4-1/40
Z 33005/41/40
As in the last row, all the coefficients are positive, then the halt condition is obey, getting the optimal solution.
The optimal solution is given by the Z value, at the column of the solution values, in this case: 33. In the same column, you can notice the point where it is reached, watching the rows correspondents to the decision variables that come at base:(x,y) = (3,12)




Thursday, 19 July 2012

Logarithms

Logarithms

 In its simplest form, a logarithm answers the question:


How many of one number do we multiply to get another number?

Example

How many 2s do we multiply to get 8?

Answer: 2 × 2 × 2 = 8, so we needed to multiply 3 of the 2s to get 8

So the logarithm is 3


So these two things are the same:
logarithm concept 

2 cubed 


 


Negative Logarithms

A negative logarithm means how many times to divide by the number. 

Example: What is log5(0.008) ... ?
1 ÷ 5 ÷ 5 ÷ 5 = 5-3, so log5(0.008) = -3


Number How Many 10s Base-10 Logarithm
larger-smaller .. etc..
1000 1 × 10 × 10 × 10 log10(1000) = 3
100 1 × 10 × 10 log10(100) = 2
10 1 × 10 log10(10) = 1
1 1 log10(1) = 0
0.1 1 ÷ 10 log10(0.1) = -1
0.01 1 ÷ 10 ÷ 10 log10(0.01) = -2
0.001 1 ÷ 10 ÷ 10 ÷ 10 log10(0.001) = -3
.. etc..

The Word

"Logarithm" is a word made up by Scottish mathematician John Napier (1550-1617), from the Greek word logos meaning "proportion, ratio or word" and arithmos meaning "number", ... which together makes "ratio-number" !




—The Relationship Animated—
                       The Relationship


 

The graph of the logarithm function logb(x) (blue)
 is obtained by reflecting  the graph of the 
function bx (red) at the diagonal line (x = y).

 
The graph of the natural logarithm (green) and 
its tangent at x = 1.5 (black) 

The natural logarithm of t agrees with the integral of 1/x dx from 1 to t:


A hyperbola with part of the area underneath shaded in grey. 
The natural logarithm of t is the shaded area 
underneath the graph of the function
 f(x) = 1/x (reciprocal of x).

                                                                         

Thursday, 12 July 2012

History of Stem and Leaf Diagram

History of Stem and Leaf Diagram

 

The Stem and Leaf plot was first used by John Tukey around 1977 in a study of volcanos.

John Tukey was a mathematician and statistician who is credited with first using the term software to describe the programs that run on computers.

Ability to organize data for analysis is important.



Wednesday, 4 July 2012

List of Planar Symmetry Groups

List of Planar Symmetry Groups

  Classes of discrete planar symmetry groups.  

The symmetry groups are named here by three naming schemes: International notation, orbifold notation, and Coxeter's bracket notation.

There are three kinds of symmetry groups of the plane:

Rosette groups

There are two families of discrete two-dimensional point groups, and they are specified with parameter n, which is the order of the group of the rotations in the group.
Family Intl
(orbifold)
Geo
[1]
Schönflies Coxeter Order Example
Cyclic symmetry n
(nn)
n Cn [n]+ n Flag of Hong Kong.svg
5-fold rotation
Dihedral symmetry nm
(*nn)
n Dn [n] 2n Topological Rose with mirrors.png
4-fold reflection

Frieze groups

The 7 frieze groups, the two-dimensional line groups, with a direction of periodicity are given with five notational names. The Schönflies notation is given as infinite limits of 7 dihedral groups. The yellow regions represent the infinite fundamental domain in each. Simple example images are given as periodic tilings on a cylinder with a periodicity of 6.

[∞,1], CDel node.pngCDel infin.pngCDel node.png

IUC
(Orbifold)
Geo Schönflies Coxeter Fundamental
domain
Example
p1
(∞∞)
p1 C [∞,1]+ Frieze group 11.png Uniaxial c6.png
p1m1
(*∞∞)
p1 C∞v [∞,1] Frieze group m1.png Uniaxial c6v.png

[∞+,2], CDel node.pngCDel 2.pngCDel node.pngCDel infin.pngCDel node.png

IUC
(Orbifold)
Geo Schönflies Coxeter Fundamental
domain
Example
p11g
(∞x)
p.g1 S2∞ [∞+,2+] Frieze group 1g.png Uniaxial s6.png
p11m
(∞*)
p.1 C∞h [∞+,2] Frieze group 1m.png Uniaxial c6h.png

[∞,2], CDel node.pngCDel 2.pngCDel node.pngCDel infin.pngCDel node.png

IUC
(Orbifold)
Geo Schönflies Coxeter Fundamental
domain
Example
p2
(22∞)
p2 D [∞,2]+ Frieze group 12.png Uniaxial d6.png
p2mg
(2*∞)
p2g D∞d [∞,2+] Frieze group mg.png Uniaxial d6d.png
p2mm
(*22∞)
p2 D∞h [∞,2] Frieze group mm.png Uniaxial d6h.png

Wallpaper groups

The 17 wallpaper groups, with finite fundamental domains, are given by International notation, orbifold notation, and Coxeter notation, classified by the 5 Bravais lattices in the plane: square, oblique (parallelogrammatic), hexagonal (60 degree rhombic), rectangular, and centered rectangular (rhombic).
The p1 and p2 groups, with no reflectional symmetry, are repeated in all classes. The related pure reflectional Coxeter group are given with all classes except oblique.
Square, [4,4], CDel node.pngCDel 4.pngCDel node.pngCDel 4.pngCDel node.png
IUC
(Orbifold)
Geometric
Coxeter
Fundamental
domain
p1
(o)
p1
[∞+,2,∞+]
Wallpaper group diagram p1 square.svg
p2
(2222)
p2
[1+,4,4]+
Wallpaper group diagram p2 square.svg
p2gg
pgg
(22x)
pg2g
[4+,4+]
Wallpaper group diagram pgg square2.svg
Wallpaper group diagram pgg square.svg
p2mm
pmm
(*2222)
p2
[1+,4,4]
Wallpaper group diagram pmm square.svg
c2mm
cmm
(2*22)
c2
[[4+,4+]]
Wallpaper group diagram cmm square.svg
p4
(442)
p4
[4,4]+
Wallpaper group diagram p4 square.svg
p4gm
p4g
(4*2)
pg4
[4+,4]
Wallpaper group diagram p4g square.svg
p4mm
p4m
(*442)
p4
[4,4]
Wallpaper group diagram p4m square.svg
Parallelogrammatic (oblique)
p1
(o)
p1
[∞+,2,∞+]
Wallpaper group diagram p1.svg
p2
(2222)
p2
[∞,2,∞]+
Wallpaper group diagram p2.svg
Hexagonal [6,3], CDel node.pngCDel 6.pngCDel node.pngCDel 3.pngCDel node.png
IUC
(Orbifold)
Geometric
Coxeter
Fundamental
domain
p1
(o)
p1
[∞+,2,∞+]
Wallpaper group diagram p1 half.svg
p2
(2222)
p2
[∞,2,∞]+
Wallpaper group diagram p2 half.svg
c2mm
cmm
(2*22)
c2
[∞,2+,∞]
Wallpaper group diagram cmm half.svg
p3
(333)
p3
[1+,6,3+]
Wallpaper group diagram p3.svg
p3m1
(*333)
p3
[1+,6,3]
Wallpaper group diagram p3m1.svg
p31m
(3*3)
h3
[6,3+]
Wallpaper group diagram p31m.svg
p6
(632)
p6
[6,3]+
Wallpaper group diagram p6.svg
p6mm
p6m
(*632)
p6
[6,3]
Wallpaper group diagram p6m.svg
Hexagonal [3[3]], CDel node.pngCDel split1.pngCDel branch.png
p3
(333)
p3
[3[3]]+
Wallpaper group diagram p3.svg
p3m1
(*333)
p3
[3[3]]
Wallpaper group diagram p3m1.svg
p31m
(3*3)
h3
[3[3[3]]+]
Wallpaper group diagram p31m.svg
p6
(632)
p6
[3[3[3]]]+
Wallpaper group diagram p6.svg
p6mm
p6m
(*632)
p6
[3[3[3]]]
Wallpaper group diagram p6m.svg

 

List of spherical symmetry groups

 

Spherical symmetry groups are also called point groups in three dimensions, however this article is limited to the finite symmetries.
There are five fundamental symmetry classes which have triangular fundamental domains: dihedral, cyclic, tetrahedral, octahedral, and icosahedral symmetry.
This article lists the groups by Schoenflies notation, Coxeter notation[1], orbifold notation[2], and order. John Conway uses a variation of the Schoenflies notation, named by one or two upper case letters, and whole number subscripts. The group order is defined as the subscript, unless the order is doubled for symbols with a plus or minus, "±", prefix.[3]
Hermann–Mauguin notation (International notation) is also given. The crystallography groups, 32 in total, are a subset with element orders 2, 3, 4 and 6.

Involutional symmetry

There are four involutional groups: no symmetry, reflection symmetry, 2-fold rotational symmetry, and central point symmetry.
Intl Geo
[5]
Orbifold Schönflies Conway Coxeter Order Fundamental
domain
1 1 1 C1 C1 [ ]+ 1 Sphere symmetry group c1.png
2 2 22 D1
= C2
D2
= C2
[2]+ 2 Sphere symmetry group c2.png
Intl Geo Orbifold Schönflies Conway Coxeter Order Fundamental
domain
1 22 × Ci
= S2
CC2 [2+,2+] 2 Sphere symmetry group ci.png
2
= m
1 * Cs
= C1v
= C1h
±C1
= CD2
[ ] 2 Sphere symmetry group cs.png

Cyclic symmetry

There are four infinite cyclic symmetry families, with n=2 or higher. (n may be 1 as a special case)
Intl Geo Orbifold Schönflies Conway Coxeter Order Fundamental
domain
2 2 22 C2
= D1
C2
= D2
[2]+ 2 Sphere symmetry group c2.png
mm2 2 *22 C2v
= D1h
CD4
= DD4
[2] 4 Sphere symmetry group c2v.png
4 42 S4 CC4 [2+,4+] 4 Sphere symmetry group s4.png
2/m 22 2* C2h
= D1d
±C2
= ±D2
[2,2+] 4 Sphere symmetry group c2h.png
Intl Geo Orbifold Schönflies Conway Coxeter Order Fundamental
domain
3
4
5
6
n
3
4
5
6
n
33
44
55
66
nn
C3
C4
C5
C6
Cn
C3
C4
C5
C6
Cn
[3]+
[4]+
[5]+
[6]+
[n]+
3
4
5
6
n
Sphere symmetry group c3.png
3m
4mm
5m
6mm
-
3
4
5
6
n
*33
*44
*55
*66
*nn
C3v
C4v
C5v
C6v
Cnv
CD6
CD8
CD10
CD12
CD2n
[3]
[4]
[5]
[6]
[n]
6
8
10
12
2n
Sphere symmetry group c3v.png
3
8
5
12
-
62
82
10.2
12.2
2n.2




S6
S8
S10
S12
S2n
±C3
CC8
±C5
CC12
CC2n / ±Cn
[2+,6+]
[2+,8+]
[2+,10+]
[2+,12+]
[2+,2n+]
6
8
10
12
2n
Sphere symmetry group s6.png
3/m
4/m
5/m
6/m
n/m
32
42
52
62
n2
3*
4*
5*
6*
n*
C3h
C4h
C5h
C6h
Cnh
CC6
±C4
CC10
±C6
±Cn / CC2n
[2,3+]
[2,4+]
[2,5+]
[2,6+]
[2,n+]
6
8
10
12
2n
Sphere symmetry group c3h.png

Dihedral symmetry

There are three infinite dihedral symmetry families, with n as 2 or higher. (n may be 1 as a special case)
Intl Geo Orbifold Schönflies Conway Coxeter Order Fundamental
domain
222 2.2 222 D2 D4 [2,2]+ 4 Sphere symmetry group d2.png
42m 42 2*2 D2d DD8 [2+,4] 8 Sphere symmetry group d2d.png
mmm 22 *222 D2h ±D4 [2,2] 8 Sphere symmetry group d2h.png
Intl Geo Orbifold Schönflies Conway Coxeter Order Fundamental
domain
32
422
52
622
3.2
4.2
5.2
6.2
n.2
223
224
225
226
22n
D3
D4
D5
D6
Dn
D6
D8
D10
D12
D2n
[2,3]+
[2,4]+
[2,5]+
[2,6]+
[2,n]+
6
8
10
12
2n
Sphere symmetry group d3.png
3m
82m
5m
12.2m
62
82
10.2
12.2
n2
2*3
2*4
2*5
2*6
2*n
D3d
D4d
D5d
D6d
Dnd
±D6
DD16
±D10
DD24
DD4n / ±D2n
[2+,6]
[2+,8]
[2+,10]
[2+,12]
[2+,2n]
12
16
20
24
4n
Sphere symmetry group d3d.png
6m2
4/mmm
10m2
6/mmm
32
42
52
62
n2
*223
*224
*225
*226
*22n
D3h
D4h
D5h
D6h
Dnh
DD12
±D8
DD20
±D12
±D2n / DD4n
[2,3]
[2,4]
[2,5]
[2,6]
[2,n]
12
16
20
24
4n
Sphere symmetry group d3h.png

Polyhedral symmetry

There are three types of polyhedral symmetry: tetrahedral symmetry, octahedral symmetry, and icosahedral symmetry, named after the triangle-faced regular polyhedra with these symmetries.
[3,3]
Intl Geo Orbifold Schönflies Conway Coxeter Order Fundamental
domain
23 3.3 332 T T [3,3]+
= [3+,4,1+]
12 Sphere symmetry group t.png
m3 43 3*2 Th ±T [3+,4]
= [[3,3]+]
24 Sphere symmetry group th.png
43m 33 *332 Td TO [3,3]
= [3,4,1+]
24 Sphere symmetry group td.png
[3,4]
Intl Geo Orbifold Schönflies Conway Coxeter Order Fundamental
domain
432 4.3 432 O O [3,4]+
= [[3,3]]+
24 Sphere symmetry group o.svg
m3m 43 *432 Oh ±O [3,4]
= [[3,3]]
48 Sphere symmetry group oh.png
[3,5]
Intl Geo Orbifold Schönflies Conway Coxeter Order Fundamental
domain
532 5.3 532 I I [3,5]+ 60 Sphere symmetry group i.png
532/m 53 *532 Ih ±I [3,5] 120 Sphere symmetry group ih.png