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