CHESTAHEDRON (3D solid: 7-faced, equal-area, "heptahedron")
- Author:
- swami_mathtraveler
INTRO TO HISTORY/MATH OF THE CHESTAHEDRON
The “Chestahedron” (heptahedron) was artistically and mathematically discovered by “artist/sculptor/geometrician” and teacher, Frank Chester, and revealed circa 2000. He has shared it extensively, with particular emphasis on the discovery process, and derivative forms, plus the “sacred” nature of such. Just google “Chestahedron” [web][images].
The solid is a special case of the diminished trigonal trapezohedron; this one having 7 faces of equal area. The math has been worked out and posted in a nice spreadsheet by Dr. Karl Maret. (And I have deciphered/implemented the trigonometry.)
- Launch page for the spreadsheet here.
- Direct link here.
- Nice summary in extensive article here.
(Note: I have set up the worksheet to coincide fairly closely with Dr. Maret’s notation, etc.)
WORKSHEET DETAILS
This Chestahedron is fully wired! The default configuration sets the “magic” parameters to realize the equal area condition. These constraints can be loosened to allow exploration of derivative forms. Angles, lengths, constraints, etc. can be reset using various controls. Selective display controls are provided to explore the underlying details.
So, take if for a spin!…
NAVIGATION/INTERACTION TIPS
’PC(Mac)’
- 3D Orbit: R-click(2-finger)/drag;
Spin by swiping (great mode to see structure).
While spinning, other controls can be activated/manipulated.
- Zoom: 2-finger glide.
- Pan: Sh+Click.
- Slider resolution: Click then adjust w/ cursor for “normal” res; combine w/ Alt(Option) for “coarse”; and Sh for “fine”.
- Full reset: Click double-arrow circle icon at top-center (in web version).
SOME FEATURES
- ’Graph it!’: WIP! Attempting to work around issues in GeoGebra, and in Desmos:( Links to image of graph here; and Desmos here. *Added a second graph!: 3D surface plot. Use slider to switch between graphs.
- ’Antiprism’: "a polyhedron composed of two parallel copies of some particular n-sided polygon, connected by an alternating band of triangles."
- ’Stellation!’: Extend edges +/- faces to generate additional geometry. These look like star points, or "stellar".
- ’Net!’: For details, click the text below the 'Net' checkbox. Basically, it's a 3D solid unfolded flat, like a cardboard box, for example.
- ’Dual polyhedron!’: Several methods available to determine the derived form. Details to follow... FYI, this feature is still sorta Alpha. (I.e. still crashes under certain conditions. Working on fixing things, but in the meantime it's good enough to have a fun play with :))
- ’Construction Objects’: These relate to constructing the duals. Planes are made based on each polyhedron vert and a related ray. These planes intersect and form lines. These lines intersect and form points. These points are connected to make the faces of the dual.
PROPOSAL FOR CHESTAHEDRON VARIANTS: “k-Chestahedra”
- Allow kite area/triangle area ratios other than 1:1.
- Let k be a positive, non-zero integer representing the areas ratio.
- This defines a family of “k-Chestahedra”.
* Discover these by playing with the worksheet…
OTHER RELATED LINKS
- Make a paper one!
- A forum discussion on the object.
- More?...