Carved from geologic erosion, there are three remaining natural bridges of white sandstone in the National Bridges National Monument in Utah:  the Hopi names are Kachina, Owachomo, and Sipapu.  On Brougham Bridge, Dublin there is a commemorative plaque dedicated to the Irish mathematician, William Hamilton, who discovered quaternions as he crossed the bridge over the Royal Canal, on his way to the Irish Academy in 1843.

The inscription reads Here as he walked by on the 16th of October 1843 Sir William Rowan Hamilton in a flash of genius discovered the fundamental formula for quaternion multiplication  i2  =  j2  =  k2  =   i j k  = – 1  and cut it on a stone of this bridge.

Quaternions are a type of hypercomplex number that can describe a four dimensional vector field as spacetime, written as this linear combination:          H = a 1 + b i + c j + d k where a, b, c, d  are real numbers and i, j, k, are three imaginary units.  These complex numbers can represent rotations in three-dimensional and four-dimensional space

Multiplication follows a cyclic pattern:                                     i  j = k ;    j  k = i ;    k  i = j 

Reverse the order, and products are negative units:      i  k = – j  ;    k  j = – i  ;    j  i = – k

quaternion picBecause the order of multiplication matters, quaternions (and three-dimensional rotations) are not commutative.  To be specific, the quaternions are the smallest non-Abelian mathematical group with all its subgroups normal – its order is 8.

a mathematical group is a special relationship among the elements in a set with the following rules:

      there must be closure:

after a binary operation, the result is still in the original set of elements – for example,      given elements A, B, in set G, the product AB must also be in G

     the elements must be associative:

for A, B, and C in set G, their products remain equal after different grouping (AB)C = A(BC)

     there must exist an identity element, I,  in the set, such that IA = AI = A. 

for addition, the identity is zero, for multiplication the identity is one.

     there must exist an inverse of each element:

for each element A in G, there is an element B = A-1 (the inverse), such that AA-1 = A-1A = I (the identity).  For example, the sum of 3 and its additive inverse, -3, is the additive identity, zero, and the product of 2 and its inverse, ½ , is the multiplicative identity, one.

The four units and eight elements in the quaternion group:  { 1, -1, i, –i, j, –j, k, –k }

If i = a, j = b, and I = 1, a and b generate the entire set:   { I, a, b, ab, ba, a2, a3, b3 }

Thus, all four rules are satisfied with the quaternion set:  closure, associativity, an identity, and inverses for each element.

The two generators, i and j, their inverses, and all elements in the set can be described visually with a Cayley diagram, named after the 19th century British mathematician, Arthur Cayley.  In general, the graph is a geometric depiction of a group as a network with directed edges connecting vertices – these segments represent multiplication by the group generators and their inverses, yielding all of the set’s elements.

For this quaternion diagram, i is a red line, and j is a green one:

cayley diagram for quaternions

Multiplying in the direction of the arrow, away from a vertex, is the same as multiplying by either i or j, and their product is the next vertex.  Multiplying in the opposite direction is the same as multiplying a vertex by either –i or –j, and their product is the previous vertex.

For example, to multiply:          k j 

find vertex k, use the green line in the direction away from the vertex to represent j, and find their product = –i

The same result could be found algebraically, by substituting the previous identities:

Since k= i j, for k j we have ( i j ) j = i ( j2 ), and since j2 = -1, we have i ( -1 ) = –i

   therefore, k j = –i

Another example, to multiply  j ( –i )

find vertex j, use the red line in the opposite direction to represent –i their product is k

Again, an algebraic solution is also possible:

Since –i = k j, for j ( –i ) we have j ( k j ) = ( j k ) j, and since j k = i, we have ( i ) j

Because  i j = k, we have:     j ( – i ) = k

Furthermore, the product of any finite string of these elements, no matter how many, will always result in one of the 8 quaternion members; their specific order represents a set of directions for moving along the path of this network.


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