On the page Some Theorems on Rotation and Translation, there it covers the product of rotations.

I have learned, that with Complex Numbers, these theorems become trivial to prove, and one can obtain the equivalent rotation or translation much easily. (from Visual Complex Analysis , by Tristan Needham. p.17. Buy at amazon )

Complex Numbers have interesting history. Here we will only give a formal style definition, but with some explanations on why they are so defined.

Complex Number is defined as a pair {a,b}, where a and b are real numbers.

Addition of complex number is defined as: {a,b} ⊕ {c,d} := {a + c, b + d}

Multiplication is defined as: {a, b}⊗{c, d} := {a*c - b*d, b*c + a*d}

This definition of multiplication has important meaning when interpreted geometrically. Complex numbers can be thought as vectors, with a multiplication rule.

Here's a explanation: Suppose the distance from {c,d} to {0,0} is r, and suppose it makes an angle θ with the positive x-axis. Then, a complex number {a,b} multipied by {c,d} by the definition {a*c - b*d, b*c + a*d}, is geometrically equivalent of rotating the point {a,b} by θ, then scale it by r.

If A and B are complex numbers, than A⊗B == B⊗A. The order does not matter.

A rotation of θ around origin is expressed in complex number as multiplication by {Cos[θ],Sin[θ]}. In other words, a point {a,b} rotated by θ around the origin can be written as {a,b}⊗{Cos[θ],Sin[θ]}.

A translation by {a,b} is expressed as adding a complex number {a,b}.

Here's how to find products of rotation, using complex numbers.

A rotation of θ centered on A can be done by a sequence of:

 1. translation of -A,
 2. rotation of θ on origin,
 3. translate it back by A.

In formulas, we have:

 r[A,θ] == t[A]^-1 * r[0,θ] * t[A].
Expressed as complex numbers, the right hand side is

 R (Z - A) + A
where Z is the complex number to be acted on, and R is {Cos[θ],Sin[θ]}.

Expand and collect we have

 R Z + (-R+1) A

The R Z term is a rotation of Z, while the other term is a
translation. So, we have shown that a rotation on arbitrary center is
equivalent to a rotation on origin followed by a translation.

Similarly, a translation followed by a rotation on origin:

 t[A] * r[0,θ]

written in complex numbers

 (Z + A) R

is equal to

 Z R + A R

meaning that it is equivalent to a rotation on origin followed by a

A rotation θ on a point B followed by a translation is also
equivalent to a rotation on origin followed by a translation. Witness:

r[B,θ] * t[A]

Since rotation on a arbitrary point B is equivalent to rotation on
origin followed by a translation, as show above, so we can rewrite the
r[B,θ] to be

r[{0,0},α] * t[C]

for some α and C. Thus

r[B,θ] * t[A] == r[{0,0},α] * t[C] * t[A]

This shows that a rotation on a point followed by a translation is
just a rotation on origin followed by a translation.

The exact coordinate any product of symmetry can be easily calculated
by complex numbers.

The sequence of rotations
 r[{a1,a2},α] * r[{b1,b2},β]
applied to the point

in complex numbers is then:

    (  ({x,y} - {a1,a2}) * {Cos[α],Sin[α]} + {a1,a2}  )
    - {a1,a2}
 * {Cos[β],Sin[β]}
 + {b1,b2}

In Mathematica, if we define the complex multiplication as cTimes, and
addition as cPlus:

 cPlus[{a_, b_}, {c_, d_}] := {a + c, b + d}
 cTimes[{a_, b_}, {c_, d_}] := {a*c - b*d, b*c + a*d}

then our sequence of rotations:

cPlus[cTimes[cPlus[cPlus[cTimes[cPlus[{x, y}, -{a1, a2}], {Cos[α],
Sin[α]}], {a1, a2}], -{b1, b2}], {Cos[β], Sin[β]}], {b1,

Expands to:

{b1 + Cos[β]*(a1 - b1 + (-a1 + x)*Cos[α] - (-a2 +
y)*Sin[α]) - (a2 - b2 + (-a2 + y)*Cos[α] + (-a1 +
x)*Sin[α])*Sin[β], b2 + Cos[β]*(a2 - b2 + (-a2 +
y)*Cos[α] + (-a1 + x)*Sin[α]) + (a1 - b1 + (-a1 +
x)*Cos[α] - (-a2 + y)*Sin[α])*Sin[β]}

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Wallpaper Groups

  1. Introduction
  2. Theorems on Rotation and Translation
  3. The Discontinuous Groups
  4. Derivation and Classification of Groups
  5. The 17 Wallpaper Groups
  6. Wallpaper Gallery
  7. References
  8. Addendum