actualizing mathematics: osculating circles, coved ceiling, desargues duals, penrose tiles, and...
TRANSCRIPT
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Actualizing Mathematics:
Osculating Circles, Coved Ceiling, Desargue’s Duals, Penrose Tiles, and
Szilassi Sculptures in Public and Private Gardens
Michael Lachance
Professor of Mathematics
UM-Dearborn
March 2009
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• Tiling– Penrose Tile Project
• Blacksmithing– Domed Ceiling – Szalassi Polyhedron
• Landscaping– Osculating Circles– Elliptical Stairs– Desargue’s Theorems
Present and Future Projects
Domed Ceiling Szalassi Polyhedron Osculating Circle Elliptical Stair Penrose Tile Desargue’s Theorem
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Penrose Tiles
Domed Ceiling Szalassi Polyhedron Osculating Circle Elliptical Stair Penrose Tile Desargue’s Theorem
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UM-Dearborn Alumni Wall Mosaic
Domed Ceiling Szalassi Polyhedron Osculating Circle Elliptical Stair Penrose Tile Desargue’s Theorem
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Special Figures, Empires
Domed Ceiling Szalassi Polyhedron Osculating Circle Elliptical Stair Penrose Tile Desargue’s Theorem
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Penrose Tiles in Public Places
Domed Ceiling Szalassi Polyhedron Osculating Circle Elliptical Stair Penrose Tile Desargue’s Theorem
Bachelor Hall, Miami University22 ft in diameter,tiles 10.5 x 17 inches
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Penrose Tiles in Public Places
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Domed Ceiling Szalassi Polyhedron Osculating Circle Elliptical Stair Penrose Tile Desargue’s Theorem
Carleton College15 ft in diameter
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Penrose Tiles in Private Places
Domed Ceiling Szalassi Polyhedron Osculating Circle Elliptical Stair Penrose Tile Desargue’s Theorem
Alex Feldman’s shower floor
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Penrose Tiles at UM-Dearborn
Domed Ceiling Szalassi Polyhedron Osculating Circle Elliptical Stair Penrose Tile Desargue’s Theorem
7 ft square
Custom tiles from…
Pewabic Pottery of Detroit
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Domed Ceiling Szalassi Polyhedron Osculating Circle Elliptical Stair Penrose Tile Desargue’s Theorem
Copper Foyer Dome
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What shape to cut the “wedges” from 2x4 sheets of copper?
Domed Ceiling Szalassi Polyhedron Osculating Circle Elliptical Stair Penrose Tile Desargue’s Theorem
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Domed Ceiling Szalassi Polyhedron Osculating Circle Elliptical Stair Penrose Tile Desargue’s Theorem
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• Name coined by Martin Gardiner, November 1978
• Each of its faces touches all the other faces
• Euler's formula: f+v-e=2-2h
• Seven-color conjecture
• http://www.drking.plus.com/hexagons/szilassi/index.html
Szilassi Polyhedron
faces vertices edges holes
Tetrahedron 4 4 6 0
Szilassi Polyhedron
7 14 21 1
Domed Ceiling Szalassi Polyhedron Osculating Circle Elliptical Stair Penrose Tile Desargue’s Theorem
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• Commercially available thru Hans Schepker Lighting
• A UM-Dearborn courtyard sculpture?
Public Szilassi
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Domed Ceiling Szalassi Polyhedron Osculating Circle Elliptical Stair Penrose Tile Desargue’s Theorem
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Face information
Domed Ceiling Szalassi Polyhedron Osculating Circle Elliptical Stair Penrose Tile Desargue’s Theorem
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Edges with mass
Domed Ceiling Szalassi Polyhedron Osculating Circle Elliptical Stair Penrose Tile Desargue’s Theorem
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Wall & Patio Designs
Domed Ceiling Szalassi Polyhedron Osculating Circle Elliptical Stair Penrose Tile Desargue’s Theorem
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Wall & Patio Execution
Domed Ceiling Szalassi Polyhedron Osculating Circle Elliptical Stair Penrose Tile Desargue’s Theorem
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Pendulum Phase Plane Portrait
Domed Ceiling Szalassi Polyhedron Osculating Circle Elliptical Stair Penrose Tile Desargue’s Theorem
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Osculating Circles, Decaying Sines
Domed Ceiling Szalassi Polyhedron Osculating Circle Elliptical Stair Penrose Tile Desargue’s Theorem
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Osculating & Concentric Circles
Domed Ceiling Szalassi Polyhedron Osculating Circle Elliptical Stair Penrose Tile Desargue’s Theorem
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Osculating & Concentric Circles
Domed Ceiling Szalassi Polyhedron Osculating Circle Elliptical Stair Penrose Tile Desargue’s Theorem
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Elliptical Stair
Wouldn’t this be nice?Existing
Domed Ceiling Szalassi Polyhedron Osculating Circle Elliptical Stair Penrose Tile Desargue’s Theorem
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Elliptical Stair
ProposedExisting
Domed Ceiling Szalassi Polyhedron Osculating Circle Elliptical Stair Penrose Tile Desargue’s Theorem
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Desargue’s Theorem
Domed Ceiling Szalassi Polyhedron Osculating Circle Elliptical Stair Penrose Tile Desargue’s Theorem
Let three lines lie on a common point.
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Desargue’s Theorem
Domed Ceiling Szalassi Polyhedron Osculating Circle Elliptical Stair Penrose Tile Desargue’s Theorem
Let three lines lie on a common point. Draw points A and A’ on one line, points B and B’ on a second line, and points C and C’ on a third line.
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Desargue’s Theorem
Domed Ceiling Szalassi Polyhedron Osculating Circle Elliptical Stair Penrose Tile Desargue’s Theorem
Let three lines lie on a common point. Draw points A and A’ on one line, points B and B’ on a second line, and points C and C’ on a third line. The points on the lines AB and A’B’, on the lines AC and A’C’, and on the lines BC and B’C’ …
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Desargue’s Theorem
Domed Ceiling Szalassi Polyhedron Osculating Circle Elliptical Stair Penrose Tile Desargue’s Theorem
Let three lines lie on a common point. Draw points A and A’ on one line, points B and B’ on a second line, and points C and C’ on a third line. The points on the lines AB and A’B’, on the lines AC and A’C’, and on the lines BC and B’C’ lie on a common line.
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Desargue’s Dual Theorem
Domed Ceiling Szalassi Polyhedron Osculating Circle Elliptical Stair Penrose Tile Desargue’s Theorem
Let three lines lie on a common point. Draw points A and A’ on one line, points B and B’ on a second line, and points C and C’ on a third line. The points on the lines AB and A’B’, on the lines AC and A’C’, and on the lines BC and B’C’ lie on a common line.
Let three points lie on a common line. Draw lines A and A’ thru one point, lines B and B’ thru a second point, and lines C and C’ thru a third point. The lines thru the points AB and A’B’, thru the points AC and A’C’, and thru the points BC and B’C’ lie on a common point.
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Desargue’s Dual Theorem
Domed Ceiling Szalassi Polyhedron Osculating Circle Elliptical Stair Penrose Tile Desargue’s Theorem
Let three points lie on a common line.
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Desargue’s Dual Theorem
Domed Ceiling Szalassi Polyhedron Osculating Circle Elliptical Stair Penrose Tile Desargue’s Theorem
Let three points lie on a common line. Draw lines A and A’ thru one point, lines B and B’ thru a second point, and lines C and C’ thru a third point.
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Desargue’s Dual Theorem
Domed Ceiling Szalassi Polyhedron Osculating Circle Elliptical Stair Penrose Tile Desargue’s Theorem
Let three points lie on a common line. Draw lines A and A’ thru one point, lines B and B’ thru a second point, and lines C and C’ thru a third point. The lines thru the points AB and A’B’, thru the points AC and A’C’, and thru the points BC and B’C’ …
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Desargue’s Dual Theorem
Domed Ceiling Szalassi Polyhedron Osculating Circle Elliptical Stair Penrose Tile Desargue’s Theorem
Let three points lie on a common line. Draw lines A and A’ thru one point, lines B and B’ thru a second point, and lines C and C’ thru a third point. The lines thru the points AB and A’B’, thru the points AC and A’C’, and thru the points BC and B’C’ lie on a common point.
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Desargue’s Theorem, Dwarf Orchards
Domed Ceiling Szalassi Polyhedron Osculating Circle Elliptical Stair Penrose Tile Desargue’s Theorem
• http://www.mtholyoke.edu/courses/jmorrow/Desargues%27Theorem.html
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Desargue’s Theorem, Dwarf Orchards
Domed Ceiling Szalassi Polyhedron Osculating Circle Elliptical Stair Penrose Tile Desargue’s Theorem
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“Doing” mathematic
s
QuickTime™ and a decompressor
are needed to see this picture.
Domed Ceiling Szalassi Polyhedron Osculating Circle Elliptical Stair Penrose Tile Desargue’s Theorem