Originally posted via website on Feb 13, 1997 to NCTM. This web version first created Feb 27, 1997, HTML formatting, pictures and hyperlinks added, text unchanged -- except for a fixed typo or two (including changing NCMT to NCTM) -- from the original. Pictures by Richard Hawkins and Chris Fearnley.


FR: Kirby Urner
Ad hoc committee for defining
new curriculum standards,
4D Solutions
TO: image The National Council of Teachers of Mathematics
RE: New geometry standards for the 21st century

In response to the question:

"2. What are new issues that should be addressed in an updated version of the Standards? Click to reply."

at: http://www.nctm.org/future-standards/future-questions.htm [now defunct -KU]


Kirby Urner is a former high school math teacher and math curriculum text book consultant for McGraw-Hill, now writing public domain curriculum material for the World Wide Web in collaboration with many talented individuals. He is an award-winning explorer in the new field of synergetic geometry, author of Synergetics in the 1990s (published journal), and former webmaster for the Buckminster Fuller Institute (BFI). He holds a BA from Princeton University.

What follows is:

(A) an excerpt of a recent posting to Synergetics-L, a listserv devoted to Synergetics, followed by

(B) an excerpt from a hypothetical learning program demonstrating how some of this new curriculum material might be packaged for 10th graders

For further reading, visit the 4D Solutions website, Synergetics on the Web, at:


(A) From a posting to Synergetics-L by K. Urner, 2/13/97, a letter to David Koski (another award-winning Synergetics explorer)...

How to best package mathematics, including geometry, most efficiently and accessibly, is one of those open questions which retrieves different answers with the times. I was the target of 'new math', heavy on the set theory. Although there's been considerable backlash, in my case that curriculum was right on. My parents, like many, maybe thought propositional calculus (so-called truth tables), Venn diagrams, and the unions and intersections of sets was way too abstract and newfangled to have much relevance to my future. But here I sit, writing in 'standard query language' (SQL) against sets of records, devising inner joins to retrieve their intersections, filtering by boolean-style propositions. I'm not claiming 'new math' was right for everyone, but I think the curriculum heads were brainstorming in the right direction for a lot of us.

But in 10th grade, no one explained to me that the tetrahedron inscribed in a cube has 1/3rd its volume, that the leftover scraps of cube assemble into a half-octahedron of volume 2/3rds. By shifting 'unity duty' onto the tetrahedron's shoulders [sic] for a change, we get a cube of volume 3, an octahedron of volume 4. And a rhombic dodecahedron of volume 6, space-filling to define unit-volumed couplers and packed-sphere domains, dissolving into a lattice of complementary tetrahedral and octahedral voids, the ones we've already met, of volumes 1 and 4.

cubetet.gif - 6.2 K

No one introduced me to the spin-networks generated by axially spinning these shapes, nor to the smooth transformation between the octahedron and cuboctahedron, passing through a five-fold phase along the way, picking up the icosa (and its dual), and its spin-network as well. Superimpose the VE's 25 great circles and the icosa's 31 to get LCD triangles incorporating both 4- and 5-fold angles, a common denominator for both symmetry groups.


Personally, I also like the economy of a rhombic triacontahedron with a diameter just a hairline shy of its dodecahedral containment, with a volume of 5 on the nose, its T-mods equivolumed with the As and Bs -- leaving the door wide open to your own phi-scaled modularizations.

In just three paragraphs I've covered a lot of geometry, and it wasn't hard to grasp -- none of it was beyond a 10th grader's level of literacy. With a few high powered animations and some hands-on model making, kids would have all this stowed away in no time. With this basic toolkit absorbed, it's easier to navigate in the structured worlds of virology, chemistry, crystallography, even cartography. A lot more of the curriculum finds a foothold, once the concentric hierarchy is firmly anchored. The chemistry teacher down the hall will have a much easier time discussing the allotropes of carbon with students already so well prepared.

(B) 1999 10th grade geometry CDROM and/or internet based web pages (a simulation, showing how some of the new geometry curriculum might be packaged, with traditional methods kept in view)

Tet-oct.jpg - 5.6 K[graphics not shown]


[click with mouse to view animation]


When you intersect a regular tet with its dual, inscribing both in the duo-tet cube, you get an internal octahedron as common to both, with four 1/8th tets protruding, each through the other's faces.

That common octahedron is 1/2 either tet's volume, the protrusions (4*1/8) giving the other half of a tetrahedron's total (tet=1).

The octa's edges are 1/2 those of the tet, so to elongate all of its edges by a factor of two is to increase its volume 8-fold (power rule) i.e. is to obtain a volume of 8*1/2=4.


[click with mouse to view each segment]

  1. Cube C1
  2. Inscribe tetrahedron T1 (cube's face diags)
  3. Inscribe tetrahedron T2 (opposite diags)
  4. Break off protruding tips of T1 T1A-T1D
    (volume of T1 NOT contained in T2)
  5. Each tip has volume 1/8 of T1
    Proof: edges 1/2 original tet so 1/2^3 of volume (power rule)
  6. Break out shared octahedral volume O1
    (volume shared by T1 and T2)
  7. It has a volume of 1/2
    Proof: T1 - (T1A+T1B+T1C+T1D) = 1/2
  8. Note its edges are 1/2 the T1's edge
    Proof: edges congruent with protruding tips
  9. Double its edges to equal those of T1
  10. Volume of larger oct is now 4
    Proof: edges doubled so volume multiplies by 2^3 i.e. 8 * 1/2 (Step 7) = 4

Relevant from Euclid

Diagonals of a parallelogram bisect each other (relevant to Step 5).

Which of the following are true about the volumes in this scenario?

a. T1+T2=O1
b. T1=T2
c. C1/3=T1
d. 2*(T1A+T1B+T1C+T1D)=O1
e. T1A/T1=1/4

Answers: b,c,e [click here for explanations]

Review questions:

[click here for hints]

Adding all surface angles:
a. One tetrahedron = _____ degrees
b. One cube = _____ degrees
c. One icosahedron = _____ degrees

Number of unit-radius spheres in a 9-frequency tetrahedron = ______

Frequency at which a tetrahedral sphere-packing first aquires a nuclear sphere = _______

Extra credit:

Write a 4D Logo program to draw T1 and T2 inscribed in C1, using different pen colors for each.

[click here to run example, source code not shown]


comicstrip.gif - 17.9 K
Jack Ohman, Mixed Media, The Oregonian, 9-24-98

Synergetics on the Web
maintained by Kirby Urner