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. 

MEMORANDUM 

FR:  Kirby Urner Ad hoc committee for defining new curriculum standards, 4D Solutions 
TO:  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/futurestandards/futurequestions.htm [now defunct KU] Background: Kirby Urner is a former high school math teacher and math curriculum text book consultant for McGrawHill, now writing public domain curriculum material for the World Wide Web in collaboration with many talented individuals. He is an awardwinning 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 SynergeticsL, 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: http://www.teleport.com/~pdx4d/index.html (A) From a posting to SynergeticsL by K. Urner, 2/13/97, a letter to David Koski (another awardwinning 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 (socalled 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 booleanstyle 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. 



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 handson 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)[graphics not shown] Scenario: [click with mouse to view animation] Narration: When you intersect a regular tet with its dual, inscribing both in the duotet 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 8fold (power rule) i.e. is to obtain a volume of 8*1/2=4. Storyboard: [click with mouse to view each segment]
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 Answers: b,c,e [click here for explanations] Review questions: [click here for hints] Adding all surface angles: Number of unitradius spheres in a 9frequency tetrahedron = ______ Frequency at which a tetrahedral spherepacking 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]
Jack Ohman, Mixed Media, The Oregonian, 92498 Synergetics on the Web 