actualizing mathematics: osculating circles, coved ceiling, desargues duals, penrose tiles, and...
TRANSCRIPT
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
• 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
Penrose Tiles
Domed Ceiling Szalassi Polyhedron Osculating Circle Elliptical Stair Penrose Tile Desargue’s Theorem
UM-Dearborn Alumni Wall Mosaic
Domed Ceiling Szalassi Polyhedron Osculating Circle Elliptical Stair Penrose Tile Desargue’s Theorem
Special Figures, Empires
Domed Ceiling Szalassi Polyhedron Osculating Circle Elliptical Stair Penrose Tile Desargue’s Theorem
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
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
Penrose Tiles in Private Places
Domed Ceiling Szalassi Polyhedron Osculating Circle Elliptical Stair Penrose Tile Desargue’s Theorem
Alex Feldman’s shower floor
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
Domed Ceiling Szalassi Polyhedron Osculating Circle Elliptical Stair Penrose Tile Desargue’s Theorem
Copper Foyer Dome
What shape to cut the “wedges” from 2x4 sheets of copper?
Domed Ceiling Szalassi Polyhedron Osculating Circle Elliptical Stair Penrose Tile Desargue’s Theorem
Domed Ceiling Szalassi Polyhedron Osculating Circle Elliptical Stair Penrose Tile Desargue’s Theorem
• 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
• 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
Face information
Domed Ceiling Szalassi Polyhedron Osculating Circle Elliptical Stair Penrose Tile Desargue’s Theorem
Edges with mass
Domed Ceiling Szalassi Polyhedron Osculating Circle Elliptical Stair Penrose Tile Desargue’s Theorem
Wall & Patio Designs
Domed Ceiling Szalassi Polyhedron Osculating Circle Elliptical Stair Penrose Tile Desargue’s Theorem
Wall & Patio Execution
Domed Ceiling Szalassi Polyhedron Osculating Circle Elliptical Stair Penrose Tile Desargue’s Theorem
Pendulum Phase Plane Portrait
Domed Ceiling Szalassi Polyhedron Osculating Circle Elliptical Stair Penrose Tile Desargue’s Theorem
Osculating Circles, Decaying Sines
Domed Ceiling Szalassi Polyhedron Osculating Circle Elliptical Stair Penrose Tile Desargue’s Theorem
Osculating & Concentric Circles
Domed Ceiling Szalassi Polyhedron Osculating Circle Elliptical Stair Penrose Tile Desargue’s Theorem
Osculating & Concentric Circles
Domed Ceiling Szalassi Polyhedron Osculating Circle Elliptical Stair Penrose Tile Desargue’s Theorem
Elliptical Stair
Wouldn’t this be nice?Existing
Domed Ceiling Szalassi Polyhedron Osculating Circle Elliptical Stair Penrose Tile Desargue’s Theorem
Elliptical Stair
ProposedExisting
Domed Ceiling Szalassi Polyhedron Osculating Circle Elliptical Stair Penrose Tile Desargue’s Theorem
Desargue’s Theorem
Domed Ceiling Szalassi Polyhedron Osculating Circle Elliptical Stair Penrose Tile Desargue’s Theorem
Let three lines lie on a common point.
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.
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’ …
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.
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.
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.
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.
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’ …
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.
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
Desargue’s Theorem, Dwarf Orchards
Domed Ceiling Szalassi Polyhedron Osculating Circle Elliptical Stair Penrose Tile Desargue’s Theorem
“Doing” mathematic
s
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Domed Ceiling Szalassi Polyhedron Osculating Circle Elliptical Stair Penrose Tile Desargue’s Theorem