Grünbaum, Branko

Branko Grünbaum

Department of Mathematics
University of Washington
Seattle, WA 98195

Articles & Books

Most recent email: 28 February 2018 at PM

Dear  Prof. Dr. Branko Grünbaum:

You’ve been emeritus so long enough, you can risk anything!  I have a sincere question that might appear unfounded. It  comes out of my earlier email to you and as a result of reading  Geombinatorics.


  1. The smallest units of space and time are the Planck Length and Planck Time.
  2. The first moments of space and time are the Planck Length and Planck Time.
  3. The first manifestations of space and time are only mathematical.
  4. The most simple manifestation of that space-time moment is a sphere.
  5. The stacking of spheres creates the first lattice, the first triangles and the first tetrahedrons.
  6. Projective geometries emerge.
  7. Euclidean geometries emerge.
First email: Thu, Jan 9, 2014 at 5:21 PM Small updates: February 2018

RE:  Our five key ideas listed below

Dear Prof. Dr. Branko Grünbaum:

I have inserted five key ideas just below this note.
Is this just nonsense or should we pursue it further?
Some days I think it is a new path. Other days, I
think its nonsense. Thank you.


1. The universe is mathematically very small.
Using base-2 exponential notation from the Planck Length
to the Observable Universe, there are somewhere over 202.34
base-2 notations. NASA’s Joe Kolecki and JP Luminet (Paris
Observatory) helped us with the calculation. Our work began
in our high school geometry classes when we started with
a tetrahedron and divided the edges by 2 finding the octahedron
in the middle and four tetrahedrons in each corner. Then dividing
the octahedron we found the eight tetrahedron in each face and
the six octahedron in each corner. We kept going inside until
we found the Planck Length. We then multiplied by 2 out to the
Observable Universe. Then it was easy to standardize the measurements
by just multiplying the Planck Length by 2. In somewhere over
202 notations we go from the smallest to the largest possible
measurements of a length.

2. The very small scale universe is amazingly complex.
Assuming the Planck Length is a singularity of one vertex, consider
the expansion of vertices. By the 60th notation, of course, there are
over a quintillion vertices and at 61st notation well over 2 quintillion more
vertices. Yet, it must start most simply and here we believe the work
within cellular automaton and the principles of computational equivalence
could have a great impact. It’s mathematics of the most simple. We also
believe A.N. Whithead’s point-free geometries could have applicability.

3. This little universe is readily tiled by the simplest structure.
The universe can be simply and readily tiled with the four hexagonal plates
within the octahedron and by the tetrahedral-octahedral-tetrahedral chains.

4. The universe is delightfully imperfect.
In 1959, Frank/Kaspers discerned the 7.38 degree gap with a simple construction of five tetrahedrons (seven vertices) looking a lot like the Chrysler logo. We have several icosahedron models with its 20 tetrahedrons that we call squishy geometry. We also call it quantum geometry (just in our high school) and we guess, “Perhaps here is the opening to randomness.”

5. The Planck Length as the next big thing.
Within computational automata we might just find the early rules that generate the infrastructures for things. Fermions and protons do not show up until the 66th notation or doubling. What are we to do with those first 65?

Bruce Camber

A little history:

First principles:

A student’s related science fair project:

An early working article about it all:

February 2018 links to key documents: