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The Geometryof Piles of SaltThinking Deeply About
Simple Things.University of Utah
Teacher’s Math Circle
February 4, 2008
Troy Jones
Waterford School
• “The real voyage of discovery lies not in finding new lands, but in seeing with new eyes.”
Marcel Proust, French novelist/philosopher 1871-1922
Work on handout
• Read and answer questions
• You might want to work individually, but compare solutions and methods.
• Take time to think about why the constructions yield the desired results
• Stop at the end of page 6 and don’t look ahead (so you won’t ruin the surprise)
Angle of Repose• The angle of repose is the
slope angle of granular material in a state of rest, measured from horizontal
• Table salt has an angle of repose of about 32°
Circle
• Predict what will happen when salt is poured on a circular shape.
• Perform experiment
• What mathematics justify or nullify your predictions?
• What other mathematics can you find in the experiment?
Salt Cone Modeled with Cabri 3D
Triangle
• Predict what will happen when salt is poured onto a triangular shape.
• Perform experiment
• What mathematics justify or nullify your predictions?
• What other mathematics can you find in the experiment?
Salt Tetrahedron Modeled with Cabri 3D
Triangle Salt Ridges Modeled withGeometer’s Sketchpad
A
B
C
Quadrilateral
• Predict what will happen when salt is poured onto a quadrilateral shape.
• Perform experiment
• What mathematics justify or nullify your predictions?
• What other mathematics can you find in the experiment?
Quadrilateral Salt Ridges Modeled with Cabri 3D
Quadrilateral Salt Ridges Modeled with Geometer’s Sketchpad
A
B
C
D
Circle near edge
• Predict what will happen when salt is poured near the edge of a surface with a hole cut near the edge.
• Perform experiment
• What mathematics justify or nullify your predictions?
• What other mathematics can you find in the experiment?
Parabolic Salt Ridges Modeled with Geometer’s Sketchpad
A B
C
D
Circle inside circle
• Predict what will happen when salt is poured onto a circle with a hole cut in it.
• Perform experiment
• What mathematics justify or nullify your predictions?
• What other mathematics can you find in the experiment?
Elliptical Salt Ridge Modeled with Cabri 3D
Elliptical Salt Ridge Modeled with Geometer’s Sketchpad
A
B
C
D
Hyperbola
• How would you model with salt an hyperbola ridge?.
• Perform experiment
• What mathematics justify or nullify your predictions?
• What other mathematics can you find in the experiment?
Hperbolic Salt Ridge Modeled with Geometer’s Sketchpad
A
B
C
D
Hoberman Sphere Model of Elliptical Salt Ridge with Cabri 3D
Elliptical Salt Ridge Modeled with Cones in Cabri 3D