Actualizing Mathematics: Osculating Circles, Coved Ceiling, Desargues Duals, Penrose Tiles, and...

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

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Domed Ceiling Szalassi Polyhedron Osculating Circle Elliptical Stair Penrose Tile Desargue’s Theorem