is that the pattern of interwoven (wicker-like) white lines hides the much simpler structure of the underlying patches of color that have well defined shapes. But already by reducing the size of the figure, the color regions become clearer, and so does the simpler underlying design.
To facilitate identifying the color patches , we simplify the pattern by removing the white lines (roughly) and exposing the pattern that was hidden under their clutter.
The colored patches now become clear:
Now we can consider the transformations on this figure that Schiller talks about, and which were unclear before.
The following is largely a distillation of Schiller's discussion .
One of the symmetry transformations that leave the pattern unchanged is rotations in its plane about the origin.
The pattern is unchanged under rotations of pi/2.
It has only four positions in which it appears identical.
In general transformations on a square that leave it looking the same form a group, called dihedral.
Any transformation or bunch of transformations that leave an object appearing unchanged is called a symmetry.
Any collection of symmetries forms a group, a symmetry group.
For a square we can enumerate the possible such symmtery transformations:
rotations and reflections (flips).
We can rotate a square by &plusminus; 90 180 and 270 degrees , leaving it unchanged.
We reflect a square about either diagonal (flip it diagonally)
or reflect the sq. horizontally or vertically (flip it hor. or vertically) and it remains unchanged.
Along with counting the rotation by 0 degree, which is the equivalent of no rotation, which is called an Identity transformation (meaning no-op or do-nothing transformation) , we thus have 8 possible transformations.
To repeat, These form a group called dihedral, hence Schiller refers to it by its symbol as group D4.
Though Schiller says there is only one transformation - rotation about pi - I don't see it . What i see is that
we have rotations about pi/2 not pi. in other words , yes the figure looks the same when you rotate it 180 degrees,
but it will also look the same when you rotate it ninety degrees. So here either I misunderstand Schiller (more likely)
or something gives.
Apparently he's considering a reflection about pi. yet his group representations use cos n pi/2 and sin n p/2 .
also weirdly he cites a reflection matrix rather than a rotation matrix, unless on (p. 202) he uses a rotation matrix.
With every rotation transformation applied to the figure,
we can see that there are sets of shapes that get transformed into each other with every transformation.
Each set of these identical shapes that get transformed into one another with each transformation is called a multiplet.
To see this, we have the same figure with the multiplet sets enumerated. Again these are the sets of similarly shaped objects that transform into each other under all symmetry transformations (those that leave the properties of the object invariant - here it is the pattern, or the shape - ) such as rotation.
Some of the multiplet sets are numbered, with the elements of each set given the number of that set.
Some multiplets are not numbered, but have arrows pointing at them.
Each multiplet as a whole (its entire set of elements) has the same symmetry as the overall figure.
For some of the colored shapes, the multiplet needs four objects to make up a whole multiplet (e.g.,
in the case of multiplet numbered 1 , we have identified its four elements).
Another multiplet, number 5, looks like it has 8 elements, but I am guessing since the symmetry degree of the multiplets is the same as that of the group.
Schiller points out another multiplet that has only one member, the central star. It is a one-element multiplet.
In any symmetry system such as this arabesque pattern, each part or component of the system is "classified" by the type of multiplet it belongs to.
Multiplets are also known as group representations.
More formally than the kindergarten chitchat above,
the representation of a group (aka symmetry group) is
an assignment of linear transformation matrix A(g) to each group element s.t.
A(g ∘ h) = A(g) ∘ A(h)
More formally still it is a homomorphism from the space of linear transformations onto the elements of the symmetry group.
Representations of unitary transformations are called unitary. Unitary transformations are matrices s.t. A*=A-1.
These have eigenvalues of norm (value?) 1 (perhaps leading to principle of gauge invariance? - afterall a gauge is a seminorm, and since norm is always one, gauge is invariant?) and mappings of unitary matrix are one-to-one.
If a matrix is not 121 it is singular, its determinant is zero, and it has no inverse transformation.
Almost all representations appearing in physics are unitary.
time evolution of physical systems is always described by the always one-to-one unitary mappings because these
map from time t-1 to t in a one-to-one fashion.
All unitary representations correspond to transformations that are one-to-one and invertible (both properties ,
equivalent to saying that determinant of the matrix is not zero and the matrix is thus non-singular.
A matrix whose det is zero is called singular as it has no inverse matrix. It cannot thus belong to a group.
Schiller then discusses reducibility of representations , identifying submultiplets , and giving rise to a classification of representations.
E.g., the pattern's symmetry group (called approximate) "has eight elements.
It has the general faithful unitary and irreducible group representations." and is an octet
Giving rise to the notation D4(The so-called "approximate symmetry group") in the given figure, denoted D_4 , has eight elemnts.)
In any case it is from the symmetry that the deduction of the "list of multiplets or representations" that describe the symmetry's "building blocks" is possible.
Apparently other symmetry groups are given for other multipets? ie the representations are reducible to singlets doublets and quartets? This is what Schiller seems to say.
Schiller notes how unlike the transformations of the tiling pattern , which were discrete, the viewpoint transformations under which the world demeures unchanged are continuous and unbounded.
Siegel's Fields also states that continuous symmetry "is one of the most fundamental and important concepts of physics."
Continuity of the xforms means their representations are continually variable, without bounds and notably are magnitudes. In other words, scalars - as opposed to vectors. They can only be scalars.
Schiller notes ominously that in contrast vectors and tensors "only scalars may take discrete values", "may be discrete observables." (p. 204)
But weren't representations just made continually variable b/c nature's symmetry transformations are continuous? sounds a bit coucou .
To make things worse Schiller states that most representations also possess direction.
So they're not scalars anymore ?
Also it is states that symmetry under change of observ. pos inst or orientation => all observables are scalars, vectors, tensors or spinors (in asc. ord. of generality).
BTW rotations form (i.e., are) an Abelian group because g ∘ h = h ∘ g.
And generally it seems that a group has higher symmetry than its subgroups; iow, a group is a "larger symmetry group" than its subgroups. Makeosa della sensa.
* * *
The b&w lines thrown on top of the layer of multiplets (the color patches or geometrically shaped regions) (that get mapped to each other when undergoing transformations like rotation) are not mere spaghetti thrown on top of the rotation group representations.
That is to say, the layer of interlocking b&w lines is not trivial. the new layer is describable - i guess - by a greater number of representations for the transformation group of this pattern. (not sure this is correct though)
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