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The paperclip snub dodecahedron is a self-supporting sculpture made from 120 paperclips. Zachary Abel designed and built it in July 2011, using 60 clips bent at their middles and 60 left intact. The result approximates the chiral Archimedean solid while keeping the individual pieces recognizable as paperclips—without glue or solder.
What the paperclip snub dodecahedron is
Abel describes the work simply as “A snub dodecahedron made from 120 paperclips.” Its design answers two practical constraints: the clips must support one another without adhesive or solder, and they must remain visibly identifiable as paperclips.
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The geometry behind the form
Faces, edges and vertices
An ideal snub dodecahedron has:
- 12 pentagonal faces
- 80 equilateral triangular faces
- 150 edges
- 60 vertices
At every vertex, one pentagon and four triangles meet. That five-face arrangement creates the dense, twisting surface associated with the solid.
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Why it is called “snub”
The snub dodecahedron is a chiral Archimedean solid. It exists in two forms that are mirror images of one another; these are the two possible handednesses of the construction. A model can therefore have a consistent left- or right-handed orientation rather than being superimposable on its mirror image by ordinary rotation.
Mathematical reference values
The Royal Society’s geometric reference identifies the solid with Coxeter symbol |2 3 5. For an ideal snub dodecahedron with edge length 2, it gives a circumscribed radius of 4.3117 and a spherical-cap angular radius of 0.2341 radians, or 13.41 degrees. Those values describe the ideal mathematical solid, not measurements of Abel’s paperclip object.
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How the 120 paperclips are used
- Start with 120 clips. Abel’s documented sculpture uses that total.
- Bend 60 clips at the middle. These altered clips provide the bent members of the framework.
- Keep 60 clips intact. The unbent clips outline the central triangles.
- Interlock the two groups into the snub-dodecahedron pattern. The pieces must reproduce the solid’s vertex arrangement—one pentagon and four triangles at each vertex—while supporting one another mechanically.
- Check the finished form. Confirm that the clips remain recognizable, no glue or solder is needed, and the structure has a consistent handedness rather than switching orientation partway around the model.
The project description does not publish a bend angle, a clip brand, or a joint-by-joint assembly diagram. Those details cannot be specified reliably from the documented description alone. In practice, the clip’s wire thickness, springiness and exact proportions will affect how easily the interlocking structure holds.
Why the sculpture is an approximation
Abel represents the ratio of a paperclip’s short segment to its long segment with e. When e equals 1, the segment lengths match the ideal snub dodecahedron. Real clips generally do not satisfy that condition.
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Changing e produces a continuous family of related forms: the family passes through an icosidodecahedron at e = 0 and approaches a rhombicosidodecahedron at e = √2. Abel notes that many paperclip brands have a long-to-short ratio near √2, so the dimensions and visual character of a finished sculpture depend strongly on the clip design you choose.
Snub dodecahedron versus icosidodecahedron
| Feature | Snub dodecahedron | Paperclip model |
|---|---|---|
| Mathematical status | Chiral Archimedean solid | Physical approximation of that solid |
| Ideal vertex arrangement | One pentagon and four triangles at each vertex | Designed to reproduce the same arrangement |
| Edge lengths | Equal in the ideal form | Unequal because a paperclip has long and short segments |
| Handedness | Two mirror-image chiral forms | Should be assembled with one consistent orientation |
| Material and construction | Abstract geometric model | 120 paperclips; 60 bent and 60 intact; no glue or solder |
| Relation to the e-family | Occurs at e = 1 | Its position depends on the selected clip’s proportions |
The icosidodecahedron is therefore not simply another name for the paperclip object. It is the limiting member at e = 0 in the proportional family Abel describes, whereas the target construction is intended to represent the snub-dodecahedron form.
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Choosing clips for a closer-looking model
- Use one consistent brand and size. Mixing shapes changes the long-to-short ratio from one part of the sculpture to another.
- Prefer clips that hold their bend. Springback can loosen the interlocking framework.
- Keep the clips recognizable. Excessive reshaping defeats one of the project’s stated design constraints.
- Expect visual variation. Two brands can produce noticeably different proportions even when both are sold as standard paperclips.
How the idea became a fabrication challenge
George Hart framed the problem in the National Museum of Mathematics’ Math Monday article dated January 31, 2011: “Paper clips also have great potential in this arena, so here is a challenge for you: the snub dodecahedron.” He described the target using the same local rule—one pentagon and four equilateral triangles at each vertex—and asked whether anyone could fabricate it from paperclips.
Related paper model
The Smithsonian National Museum of American History catalogs a separate paper model by A. Harry Wheeler. Its description specifies twelve regular pentagons and eighty equilateral triangles cut and folded from paper. It is useful for comparing the face structure of a physical polyhedron, but it is not evidence that Wheeler made Abel’s paperclip sculpture.
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Is it a true snub dodecahedron?
Topologically and visually, the sculpture is intended to realize a snub dodecahedron: it uses the characteristic face arrangement and a consistent chiral orientation. Geometrically, it is an approximation because the paperclip’s long and short segments are unequal. Calling it a paperclip snub dodecahedron accurately describes both facts—the target solid and the material compromise.
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