D-forms

Surprising new 3-D forms from flat curved shapes

"The shapes are a beautiful representation of the integration of mathematics and art, an idea both engaging and challenging." Mathematics Teacher, NCTM Jan 2011

Surprising new 3D forms from flat curved shapes. Could you make your own Squaricle, 3-Dform, Wobbler and Twisted Bean yourself?

How developable surfaces grow from a single planar sheet

Developable surfaces such as these are made from a single sheet. You start with two planar shapes and join them at their perimeters: the paper bends into a smooth surface without stretching or tearing. Draw a straight line across a planar template before joining: once the paper curves, it still traces the shortest path, a geodesic, on the finished form.

The difference lies in the joining, not in folding along straight creases. The bending happens along curved edges, so curvature appears where you would not predict it. Mathematically, developable surfaces belong to the family of ruled surfaces, which links the models to real geometry.

What the D-forms book contains

• Ten models that introduce D-forms, including the Squaricle, 3-Dform, Wobbler and Twisted Bean
• Explanatory material to develop your understanding of the geometry behind each form
• Templates to explore ellipses with different aspect ratios, but all one perimeter

Equal-perimeter templates: change the shape of the ellipse, keep the edge length the same, and compare the freeform results side by side.

Curved paper modelling for mathematicians, designers and artists

Such explorations provide fascinating projects for mathematicians, designers, artists and paper engineers. Anyone interested in surface geometry gets a physical model to hold, turn and question.

"The basic idea, though simple, seems to be rather novel. It is not addressed by the standard mathematics or modelling literature. Apparently, there are no known closed form solutions that will predict what 3D shape will result when simple 2D figures are joined at their perimeter into a D-Form - raising a challenge for some serious mathematical research....Fortunately no math or computer is required to build all of the D-Forms in this book....[just] good, educational entertainment." Carlo H Sequin, Journal of Mathematics and the Arts vol 3, no 4.

In practice, this means no software and no digital mesh, unlike computer modelling of curved surfaces, where a mesh of flat triangles approximates each curve. Paper, scissors and the printed templates are enough: the bending of the sheet settles the shape.

For designers, the models also act as a physical check on a digital mesh. Developable surfaces can be built from flat sheet material, whereas a doubly curved form cannot without stretching. Paper engineers can then compare them with buckled developable surfaces, where the sheet creases or wrinkles as it bends.

Teaching geometry and visual-spatial skills with D-forms

For teachers, Mathematics Teacher, NCTM Jan 2011 comments "Classroom teachers can quickly implement D-forms using the models included in the book. The hands-on activities will help students understand simple geometrical terms (length, perimeter, two-dimensional planar shapes, and three dimensional figures). However, the greatest benefit of the activity is the potential to strengthen visual spatial skills."

The book is best suited to secondary students and adult makers who can cut and join edges accurately. Hands-on, the idea clicks when students predict a form's curvature and then test it, making their geometry reasoning visible.

A good first activity: cut two ellipses with the same perimeter, join their edges and watch a D-form take shape in your hands.