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 A Unifying Concept Across Curricula
 Geometry, music and language arts share many concepts of symmetry.

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 1 a obsolete : mutual relationship of parts (as in size, arrangement, or
measurements) : PROPORTION b :
due or balanced proportions : beauty of form or arrangement arising from
balanced proportions [with order, symmetry, and taste unblest Robert
Burns]
 2 : correspondence in size, shape, and relative position of parts that
are on opposite sides of a dividing line or median plane or that are
distributed about a center or axis : an arrangement or external form (as
in a body, a design, or a grouping) marked by bilateral conformity or
geometrical regularity
 4 : the property of a crystal of having two or more directions that are
alike in physical and crystallographic respects because of identity of
atomic structure in the directions concerned or mirrorimage relations
along such directions

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 A symmetry is a onetoone correspondence of form under some
transformation, often on opposite sides of a dividing line or plane or
about a center or an axis. This correspondence may be a
congruence, identity or equivalence.

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 An object is symmetric with respect to a given mathematical operation,
if, when applied to the object, this operation does not change the
object or its appearance. Two objects are symmetric to each other with
respect to a given group of operations if one is obtained from the other
by some of the operations (and vice versa).
 In 2D geometry the main kinds of symmetry of interest are with respect
to the basic Euclidean plane isometries: translations, rotations, reflections,
and glide reflections.

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 The idea of symmetry is intimately bound up with harmony and balance;
rhythm and repetition  essential ingredients of beauty.
 A mathematician might represent the symmetry group C_{3} as {e^{2}^{π}^{ik/}^{3
} k = 0,1,2}
 {I, R, R^{1}}
 A more elaborate
visual representation:

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 Bounded: rosettes and pinwheels.
 Linear: Strip (frieze) patterns
 Covering: wallpaper symmetries

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 11: No congruence except by horizontal translation.
 1m: Reflection in the horizontal line.
 m1: Reflection in the vertical line.
 12: Halfturn or twofold rotation.
 mm: Reflections in vertical and horizontal lines.
 mg: Reflection in vertical line; glide reflection in horizontal.
 1g: Glide reflection in the horizontal line.

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 A second instance is the paper doll pattern. Here, there are two
different fold lines. You make paper dolls by folding a strip of paper
zigzag, and then cutting out half a person. The halfperson is enough
to reconstruct the whole pattern. The quotient orbifold is a
halfperson, with two mirror lines.

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 Each 2 shows the presence of a 2fold rotation.
 * shows the presence of reflection.
 If a 2 comes after the asterisk, then the rotation centre is at the
intersection of mirror lines.
 x shows the presence of glide reflection.

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 Wikipedia’s frieze groups site uses more iconographic images to
represent these.
Which is 11, 12, 1m, m1, 1g, mg or mm?
 Here is an algorithm for deciding:

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 A tessellation or tiling is a covering of space with nonoverlapping
figures.
 A regular tessellation is a tiling with regular polygons (in 2D),
regular polyhedra (3D), or polytopes (nD).

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 Ten cyclic/dihedral types in first row.
 Crystallographic notation in second row.
 Conway notation in third row.
 Note 7 frieze patterns must be in D1, C2 or D2.

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 If we allow edgetoedge tilings that use various regular polygon tiles
then the situation becomes more interesting.
 Johannes Kepler considered semiregular tilings in his Harmonices Mundi
of 1619.

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 The angle at each vertex of an ngon is
 Thus it is easy to check
by simple arithmetic that only 17 choices of regular polygons can be
fitted around a single vertex so as to cover a neighborhood of the
vertex without gaps. What sums of
multiples of 90, 108, 120, etc. (the interior angles of regular ngons)
add up to 360?
We call each such
choice the species of the vertex, of which 17 are possible. In four of the species there are two
distinct ways in which the polygons in question may be arranged around a
vertex; the mere reversal of cyclic order is not counted as distinct.
Hence there are 21 possible types of vertices.

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 A systematic tiling that uses a mix of regular polygons with different
numbers of sides but in which all vertex figures are alike—the same
polygons in the same order—is called a semiregular tiling.
It is said that the 11 such tilings were originally identified by
Johannes Kepler – so (naturally) they’re called the Archimedean tilings.

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 In addition to the 3 regular tessellations, there are 8 more meeting the
“same polygons in the same order” condition.

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 Coincidentally, the limitation to a single type of vertex leads to a
single type of tiling, but just barely: the tilings of type (3^{4},
6) are of two mirrorsymmetric (enantiomorphic) forms that are generally
not counted as distinct.

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 Main Entry: enantio
Function: combining form
Etymology: New Latin,
from Greek, from enantios, from enanti in the presence of, from en
in + anti against
1 : opposite *enantiotropy
2 : antagonistic *enantiobiosis

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 That is, all vertices are equivalent under the symmetries of the tiling
(isometries).
 In other words, for each pair of vertices A and B it is possible to find
a motion of the plane, or a motion combined with a reflection in a line,
that carries the tiling onto itself and maps A onto B.
 These are called uniform tilings.

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 Any triangle can be thought of as half a parallelogram and so all
triangles (just as all parallelograms) tile the plane.
 More surprisingly, any quadrilateral will tile the plane. See this Sketchpad site to
demonstrate.

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 One pair of opposite parallel, congruent sides.
 Angles of 120 alternating through interior.

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 A problem for anyone to contribute to:
a survey of the growing but incomplete
story of pentagonal tilings of the plane

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 There are four consecutive points A, B, C, D on the boundary such that
 The boundary from A to B is congruent by translation to the boundary
from D to C, and
 The boundary from B to C is congruent by translation to the boundary
from A to D.

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 There are six consecutive points A, B, C, D, E, and F on the boundary
such that the boundary parts AB, BC, and CD are congruent by
translation, respectively, to the boundary parts ED, FE, and AF.
 This is an idea that Escher developed.

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 The Poincaré disk is a model for hyperbolic geometry in which a line is
represented as an arc of a circle whose ends are perpendicular to the
disk's boundary (and diameters are also permitted). Two arcs which do
not meet correspond to parallel rays, arcs which meet orthogonally
correspond to perpendicular lines, and arcs which meet on the boundary
are a pair of limits rays.

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 Also, here’s an applet illustrating that for any two points on the
interior, there is a unique perpendicular arc passing through them.

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 Complementary devils and angels fill the hyperbolic plane.
 There are (say, in the middle) kaleidescopic points of order 3 with
three mirror lines at angles of 60 degrees to one another.
 There are other points where 8 wingtips meet. Four angels’ and 4 devils’ wings. These are gyration points with
rotational symmetry, but no mirror lines. At each of these, one of 4 devils is
facing away. Conway says this is
designed to distinguish one symmetry group from another. Compare this
hyperbolic tessellation with the next:

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 Inversion of Audio Pitch When
pitch is mirrored across a point (fixed pitch) , there is an obvious
result. Up is now down, and down in now up. In that respect a scale
sounds different. But another thing happens. The mood of the scale or
chord changes.
 Let's take the first five notes of a C Major scale. When mirrored, you
end up with the first five notes of a A minor scale, descending. A happy
tune suddenly becomes sad. If you start with an A minor scale, you'll
end up with a C Major. However, if you play the *whole* scale, you'll
end up with a a Phrygian scale, coming from major, or a Mixolydian
coming from a minor scale. You get something bizarre and interesting if
you mirror a harmonic minor scale... ("harmonic major"
perhaps?)
 The same is true for chords. If you start with a C minor chord, you end
up with an A Major chord. A major 7th chord ends up as a diminished
chord with a natural 7th. Diminished and Augmented chords (having equal
intervals between each note) have the same quality when mirrored.

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 http://jan.ucc.nau.edu/~tas3/wtc/ii21.html
 Symmetry has been used as a formal constraint by many composers, such as
the arch form (ABCBA) used by Steve Reich, Béla Bartók, and James Tenney
(or swell). In classical music, Bach used the symmetry concepts of
permutation and invariance; see (external link "Fugue No. 21,"
pdf or Shockwave).

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 The black/white piano key
pattern is reflected about a vertical line of symmetry.

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 http://jan.ucc.nau.edu/~tas3/wtc/ii21.html
 Relation to Cullinane’s Diamond Theory.

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 http://en.wikipedia.org/wiki/List_of_regular_polytopes
 http://www.geom.uiuc.edu/docs/reference/CRCformulas/node56.html
 http://bbs.sachina.pku.edu.cn/stat/math_world/math/t/t089.htm
 http://aleph0.clarku.edu/~djoyce/poincare/poincare.html
 http://www.math.princeton.edu/facultypapers/Conway/
 http://www.phys.uu.nl/~fmeijer/wordpress/2006/01/tuesdayafternoonphysicspoetrythesymmetryedition/

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 http://www.education.wisc.edu/edpsych/facstaff/dws/ew/

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 http://standards.nctm.org/document/eexamples/chap6/6.4/index.htm
 http://en.wikipedia.org/wiki/Tessellation
 http://library.thinkquest.org/16661/index2.html
 http://mathforum.org/sum95/suzanne/historytess.html
 http://www.cromp.com/tess/home.html
 http://mathforum.org/dynamic/onecorona/
 http://aleph0.clarku.edu/~djoyce/poincare/index.html
 http://people.hws.edu/mitchell/tilings/Part3.html
 http://en.wikipedia.org/wiki/Wythoff_symbol
 http://www.acnoumea.nc/maths/amc/polyhedr/plan_sym_.htm
 http://michaelshepperd.tripod.com/resources/groups.html

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 http://home.inreach.com/rtowle/Polytopes/polytope.html
 http://bbs.sachina.pku.edu.cn/stat/math_world/math/t/t089.htm
 http://www.borderschess.org/KTtess.htm
 http://en.wikipedia.org/wiki/List_of_regular_polytopes
 http://www.fllo.dk/index.asp?name=symmetry

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 http://www.uwgb.edu/DutchS/symmetry/symmetry.htm
 http://webmineral.com/crystall.shtml
 http://www.chem.ox.ac.uk/icl/heyes/structure_of_solids/Strucsol.html
 http://www.molecularstation.com/molecularbiologyimages/502dnapictures/

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