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Geometry concerned with questions of shape, size, relative position of figures, and the properties of space.
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Under Euclid worked from point, line, plane and space.
Geometry originated as a practical science concerned with surveying, measurements, areas, and volumes. Under Euclid worked from point, line, plane and space. In Euclid's time… … there was only one form of space. Today we distinguish between: Physical space Geometrical spaces Abstract spaces Wikipedia
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correspondence of distance between various parts of an object
Symmetry correspondence of distance between various parts of an object Wikipedia Tiling of Hyperbolic Plane
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Symmetry Area of Geometry since before Euclid
Ancient philosophers studied symmetric shapes such as circle, regular polygons, and Platonic solids Occurs in nature Incorporated into art Example M.C. Escher
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Symmetry Broader definition as of mid-1800’s
Transformation Groups - Symmetric Figures Discrete –topology Continuous – Lie Theory and Riemannian Geometry Projective Geometry - duality
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Projective Geometry
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Symmetric Figures Groups
Symmetry Operation - a mathematical operation or transformation that results in the same figure as the original figure (or its mirror image) Operations include reflection, rotation, and translation. Symmetry Operation on a figure is defined with respect to a given point (center of symmetry), line (axis of symmetry), or plane (plane of symmetry). Symmetry Group - set of all operations on a given figure that leave the figure unchanged Symmetry Groups of three-dimensional figures are of special interest because of their application in fields such as crystallography.
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Symmetry Group Motion of Figures: Translation Rotation
Mirror – vertical and horizontal Glide
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Mirror Symmetry Atu.edu 6-13
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Rotation Symmetry Atu.edu
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Symmetry of Finite Figures
Have no Translation Symmetry Mirror Rotation Reflection by mirror m1 Reflection by mirror m2 Reflection by mirror m3 Do nothing Rotation by turn Rotation by turn
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Symmetry of Figures With a Glide And a Translation
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Vertical Mirror Symmetry
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Horizontal Mirror Symmetry
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Vertical and Horizontal Mirrors
Rotational Symmetry = Vertical and Horizontal Mirrors
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Human Face Mirror Symmetric?
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Geoboard – construct square, rhombus, rectangle, parallelogram, kite, trapezoid or isosceles trapezoid. Complete table below.
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Frieze Patterns frieze from architecture
refers to a decorative carving or pattern that runs horizontally just below a roofline or ceiling
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Frieze Patterns also known as Border Patterns
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What are the rigid motions that preserve each pattern?
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Frieze Patterns Conway, Princeton mathematician devised the following names for the different frieze groups: 1 Hop; 2 SpinHop; 3 Jump; 4 Sidle;5 Step; 6 SpinJump; 7 SpinSidle.
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Rotate the Tires Tires One Tires Two And God said unto baby Noah
"There will be a mild increase in water level, and you must find a tire, 2 cubits in diameter, and take with you two of each stuffed animal."
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Rotate the Tires
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Rotate the Tires RRH RRV H V
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Flip the Mattress
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Flip the Mattress Motion 1 A B C D Flip the Mattress Motion 2 D C B A
In Spring - spin; in Fall - flip.
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When does 9+4 =1 ?
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Modular Arithmetic Where numbers "wrap around" upon reaching a certain value—the modulus. Our clock uses modulus 12 mod 12
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What would time be like if we had a mod 24 clock?
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What would time be like if we had a mod 6 clock?
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