A Brief History

Elements of Geometrie, published in London in 1570, is the first English translations of Euclid’s Elements. John Dee’s preface to Henry Billingsley’s Elements included pop-ups. The Thomas L. Fisher Library at the University of Toronto has one of the original copies of this text and has made pictures of the original available on the Internet.

Elements of Geometrie

Moreover Cameo Wood a student at Simon’s Rock College of Bard University posted a movie on You-tube that demonstrates the operation of these pop-ups.

Another well-known example of the use of pop-up models to illustrate solid geometry is John Lodge Cowley’s Geometry Made Easy first published in 1752. The University of Michigan’s rare books library has a copy believed to date from some time in the 1760’s.

Geometry Made Easy

More recent examples In the late 1970’s BBC-TV introduced a series of programs with mathematical themes such as Think of a Number. By 1982 Johnny Ball, the star of the series produced Johnny Ball’s Think Box a book with numerous mathematical oddities including some basic pop-up models of Platonic solids.

Paul Jackson provided hands-on instruction to popularize general pop-up techniques in 1993.

Jan Baranowski explored properties of some collapsible models of the tetrahedron, cube, octahedron, rhombic dodecahedron and cubo-octahedron in 1992.

Current mathematicians explore the topic

Hans Walser and Scott Johnson have written perhaps the most serious works in this area. In 1996 Walser shared an office with Jean Pederson at Santa Clara University. Pederson was working with Peter Hilton on a project involving extensions to the Pascal Triangle that could be modeled by a cuboctahedron. Pederson hoped to have a model that would fold flat in a briefcase to travel to lectures. Walser developed a model that interested many of Pederson’s students. One student in particular, Scott Johnson, was eager to help in the development of a number of pop-up polyhedra. Johnson and Walser have written several articles describing their models and the techniques they developed to build them. Note Johnson is a Middle School teacher in California.

The Techniques and Terms• Jackson: “a self-erecting, three-dimensional

structure, formed by the action of opening a crease”– Diagonal crease– On the crease

• Ball: Automated opening– Rubber band assist

• Walser and Johnson:– rubber bands, pull strings, and paper fastener rotation

centers; edge jumping, vertex jumping, face jumping

• Dee and Crowley– Straightforward finger lifting

Jackson

• Jackson has many models employing a variety of techniques

• He shows a number of polyhedra including two “cubes” one of which is “topless”

• He also shows a hexagonal prism.

Johnny Ball• His pop-up method is very similar to the

familiar polyhedral calendar.• Ball’s models include:

– A tetrahedron (made from an envelope)– A cube which does not pop up, but does inspire a

“flexi-rube”– An octahedron which pops up by the use of a rubber

band– A dodecahedron based on two decahedral ‘flowers”

(5 pentagons arranged around a central pentagon) placed back to back but staggered with a rubber band lacing the circumference.

– An icosahedron that does not pop-up

Walser and Johnson• Edge Jumping models

– Parallel edges• Cube• Octahedron• Icosahedron• Dodecahedron

– Edges where one lies in a plane which is perpendicular to the other line.

• Tetrahedron - with triangles that touch only at a vertex

• Spiral models- where the top and bottom layers turn relative to each other as the models collapse.

• Icosahedron• dodecahedron

Walser and Johnson

• Face-jumping (two opposite faces are pressed together to flatten the model)– Icosahedron with brass paper fastener

pivots

• Vertex-jumping (two opposite vertices are pressed together to flatten the model)– cube

Tools• Geometer’s Sketchpad for the precise

measurements• Printer that can handle card stock - preferably

in color.• Some way to cut with precision…sharp fine

scissors (I used a Craft Robo)• Some way to score the fold lines…a compass

point and ruler (I used the Robo to score the folds)

• Glue or tape.• Rubber bands, string, brass paper fasteners

Tools of Geometer’s Sketchpad to deal with the

precision.• Rotations can provide exact angle

measurements.

• Mirrors can permit mirror images of complex parts.

• Reflections simplify constructions

• Midpoints, angle bisectors, parallel lines and perpendicular lines can locate folds.

Sample models to try

Johnson - on the crease

Johnny Ball’s Octahedron

Walser and Johnson- Icosahedron with pivots

Gould - Rhombicuboctahedron

What motivated this project?

References• [1] Edward Rolf Tufte, Envisioning Information, Graphics Press:

Cheshire CT, p. 16. 1990.

• [2] http://www.math.ubc.ca/people/faculty/cass/Euclid/dee/dee.html (retrieved January 11, 2008)

• [3] Cameo Wood, http://www.youtube.com/watch?v=fT1B_4XEQMU added April 6, 2007 retrieved January 11, 2008.

• [4] John Lodge Cowley, appendix to the Elements of Euclid, University of Michigan rare books library Special Collections QA31.E88S76 C87 176.

• [5] Steinhaus, H. Mathmatical Snapshots. NY: Oxford University Press. (1969).

References continued• [6] Johnny Ball, Johnny Ball’s Think Box, Puffin Books, Penguin

Books Ltd. Middlesex, England. Chapter 10. 1982.

• [7] Struyk, A. (1956). “Three folding models of polyhedra.” Mathematics Teacher , 49, 286-288.

• [8] Trigg, C. (1972).” Collapsible models of the regular ocatahedron.” Mathematics Teacher , 531-533.

• [9] Baranowski, J. (1992, September). “Pocket Polyhedra.” Mathematic Teaching , 8-9.

• [10] Baranowski, J. (1992). Journal of Polish ATM (NiM Vol. 2). Published in Polish. Personal email April 15 and 16th 2009.

References continued• [11] Scott Johnson and Hans Walser, “Pop-up Polyhedra,” The

Mathematical Gazette, Vol. 81, No. 492. (Nov., 1997), pp. 384-380.

• [12] Hans Walser, “The Pop-up Cuboctahedron”, The College Mathematics Journal, Vol. 31, No. 2 (Mar., 2000), pp. 89-92.

• [13] Geometer’s Sketchpad, software from Key Curriculum Press, Emeryville CA.

• [14] Magnus J. Wenninger, Polyhedron Models for the Classroom, National Council of Teachers of Mathematics Washington DC. 1966.

• [15] Magnus J. Wenninger, Polyhedron Models, Cambridge University Press, Cambridge, United Kingdom. 1996 reprint of 1971 edition.