Five Variations of Five Intersecting Tetrahedra

Overview

Thomas Hull founded the field of origami wireframes (or polypolyhedra per Robert Lang) when he created the classic Five Intersecting Tetrahedra (FIT) model. FIT is comprised of five tetrahedra (aka triangular pyramids), each constructed using a unit for each of six edges, that share a center-point and all intersect without overlapping. The existence of FIT has inspired many folders to design their own wireframe models, some of which I’ve re-created in my gallery.

In this post, we explore variations of the classic FIT model. We start with a table summarizing all the variations then look at each one-by-one.

Table of variations

Year Designer Variation Transformation Units
1995 Thomas Hull - - 30
2000 Daniel Kwan FIT + Tangent Sphere break each edge in half and strip weave the open ends into a sphere 60
2002 Robert Lang K2 replace each tetrahedron with four triangles, one for each face 60
2012 Byriah Loper "K5" Twenty Interlocking Tetrahedra replace each tetrahedron with four tetrahedra 120
2012 Byriah Loper "K6" Twenty Interlocking Triangular Pyramids slice off and keep only the corners of each tetrahedron 120
2014 Meenakshi Mukerji Compound of 5 Tetrahedra eliminate all spaces between frames 60

FIT

Source: http://mars.wne.edu/~thull/fit.html

What is it about FIT that makes it so satisfying? I think it’s the thick edges and the minimum complexity for a 3D frame. Less complex models can have thicker edges and be easier to follow with the eye, but they can’t have 3D frames and are less interesting. More complex models invariably have thinner edges and require more mental visualization to appreciate.

FIT + Tangent Sphere

Source: https://www.flickr.com/photos/8303956@N08/5854815733

This 20-year-old model is a novel combination of wireframes and strip-weaving, the pairing of which neither the designer nor anyone else has explored since then. Prior to and actually while trying to design this model, Kwan designed a spikier model with larger distances between each pair of broken edges: FIT + Black Hole

K2

Source: https://langorigami.com/artwork/k2-opus-391

Lang was the first person to do a systematic analysis and enumeration of polypolyhedra. He limited his study to those polypolyhedra constructable using a single unit type. His renderings of the resulting 54 possibilities are a source of inspiration for designers and it’s common to see people refer to Lang's PPH #N.

“K5” Twenty Interlocking Tetrahedra

Source: https://www.flickr.com/photos/byriahloper/7007104719

A straight-forward application of a four-tetrahedra weaving pattern by Mancini, this model preserves the shape of the original FIT but adds modern-looking flair and visual interest as the eyes follow the criss-crossing among frames of the same color. Loper diagrammed this model in his book Mind-Blowing Modular Origami: The Art of Polyhedral Paper Folding

“K6” Twenty Interlocking Triangular Pyramids

Source: https://www.flickr.com/photos/byriahloper/7832280292

The next sequence in Loper’s “K” series, which currently goes up to K10.

Compound of 5 Tetrahedra

Source: https://www.flickr.com/photos/mmukhopadhyay/15450969436

In his diagrams for FIT, Thomas Hull posed some challenges to his reader, the most interesting being to design a compound of 5 Tetrahedra, i.e. the geometric construct he started with to design FIT. I was aware of a 2006 design by Hideaki Kawashima. Mukerji herself has a prior attempt from ~2004 and calls out previous efforts by Kamiya Satoshi, -F r o y-, and Nicolás Gajardo Henríquez. Compared to these other models, Mukerji’s is much easier to fold and has interesting, non-conventional tabs and pockets.

Other variations

  • Loper’s “K” series
    • I only covered K5 and K6 here, so you can check out K3, K4, K7, K8, K9, K10
  • Aaron Pfitzenmaier’s Variable Width FIT
  • Daniel Kwan’s Woven Seventh Stellation of the Icosahedron
    • https://www.flickr.com/photos/8303956@N08/23271368212/
    • the transformation here is

      for each tetrahedron’s triangular face, create three new edges by bisecting the existing edges until they meet in the center, then discard all original edges

    • I actually folded this but didn’t include it because I botched the color ordering