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Gwen L. Fisher
Department of Mathematics
California State Polytechnic Univ.,
San Louis Obispo

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Cayley Tables and Quilts

NCB Deposit # 48

from Broom Bridge
Dublin, Ireland

Cayley Table Pattern for Quarternion group H 
Cayley Table pattern
The quarternions group consists of 8 elements  1, -1, i, -i, j, -j, k, and -k,  with the multiplication identities of
Group equations

The labeling of  i,  j, and k is unique only up to automorphisms.  Thus the same block pattern is used for each.

The H group is also known as Q8.

Quaternion group H

Another famous pattern, a.k.a. a dihedral group, features the symmetries of a triangle.
D3  (order 6, triangle)
Cayley pattern D4
Dr. Fisher's Quilt

Fisher's quilt

Quilts are constructed by joining small "pieces" into "squares."
The identity element for this quilt is the Moon over the Moutain square.
code of quilt

As for color, each element is the same color as its corresponding negative element:  1 and -1 are both blue, i and -i are red,  j and -j are green, and k and -k are both gold and black.
code of quilt

The quilt also shows the normal subgroup { 1. -1 } and its four cosets represented by the four colors.  This in turn shows a factor group of  H  isomorphic to the Klein 4-subgroup.
Klein group

One last property that is easily seen in this representation is that the quaternions group is not commutative.  The quilt and tables do not have reflectional symmetry across the main diagonal.

table with diagonal

Cayley Table Pattern for the whole H group intersecting D4.

Quilt key

The eight elements of D4 are displayed down the left and across the top.  The patterns are as follows:

(1)  vertical line of reflection
(2)  diagonal line of reflection (top left to bottom right)
(3)  horizontal line of reflection
(4)  second diagonal line of reflection (bottom left to top right)
(5)  identity
(6)  clockwise quarter turn
(7)  half turn
(8)  counter clockwise quarter turn

The eight elements of H are labeled across the bottom and on the lower right side.

The intersection of D4 and H:

Quilt illustration
Dr. Fisher's Second Quilt

Fisher's second quilt

The whole H intersects  D4 quilt.
The identity element for both groups looks like a zero, Look here..  The quilt block for each of the four lines of reflectional symmetry contains that attribute and the blocks with quarter turn elements each have the quarter-turn rotational symmetry.

Each element in the quarternions has the same coloring as its negative element:  i  and -i are both red on orange,  j and -j are both cream on navy, and k and -k are both black on purple.

The half turn in 
D4 always corresponds with -1 in H and is a black square in the quilt.

The ordering of the elements of
D4 and H was chosen so that a cyclic subgroup K of order four forms the intersections of the two multiplication tables.   Here, K is generated by the quarter turn in D4 and by  i  in H.  The two-by-two substructure in each of D4 and H shows the cosets of the normal subgroup K of each of the two groups.

For all viewers . . . .
Viewers interested in quarternions will want to visit the NCB web page showing the Broom Bridge in Dublin, Ireland where Sir William Rowan Hamilton first formulated this group's structure.  The name "H"  honors Hamilton's highly original contributions.

"Quilts" based on Cayley table patterns is an introduction to a sophisticated topic in mathematics called Group Theory.

Fisher, Gwen L.,
The Quaternions Quilts, FOCUS, The Newsletter of the Mathematical Association of America, vol. 25 (4), January, 2005, p. 4.

For the student . . . .

Each quilt piece represents one of the objects in a Group.  If group G has 8 elements, then you will have 8 different pieces quilted together to form a 8 x 8 "square."  Similarly, the Cayley table for this group will have 64 entries. 

Cayley tables are studied in an advanced topic of mathematics named Group Theory.  Some historical background will enrich your understanding.  In past centuries, the Arab world contributed equation theory leading to the powerful tool of symbolic manipulation.  The Greeks provided proof - the axiomatic method for codifying important aspects of algebraic systems.  One of the most notable intersections of these two great mathematical approaches occurs in the theory of groups.  Briefly, Group Theory is a comprehensive analysis of the concept of symmetry.

Group Theory is an area of active research in pure mathematics that is widely applied in the sciences.  In particular, physicists, chemists and biologists use Group Theory to elucidate the structures of crystalline solids as well as isolated molecules.  Group Theory has also been applied to predict the reactivity patterns of chemical compounds.

A quilt is a plane surface.  The inquisitive student might ask, just as Gauss and Hamilton wondered:

If two-dimensional complex numbers may be represented on a plane, and can be multiplied and divided in such an exquisite way, is it not possible to try the same for three dimensions using three-dimensional vectors?

Gauss and Hamilton were never able to give a conclusive answer.  The investigation of this question inspired massive developments in modern abstract algebra extending well into the 20th century when its impossibility was finally established.

  The novice may also want to explore the connection between Cayley tables, quilt patterns and modular, or "clock" arithmetic.

Quilt at MAA Section meeting
Quilt at MAA Section meeting
Dr. Fisher explaining her quilts at the Southern California-Nevada Section of the Mathematical Association of American Spring 2005 Meeting.

The NCB thanks Dr. Fisher. 
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In addition, Dr. Fisher thanks her friends, Florence Newberger of CSU Long Beach, James Hamblin of Shippensburg University, and Anton Kaul of Cal Poly, San Luis Obispo.