Now available on Kindle

Sunday, December 31, 2017 4:58 pm PST

The Kindle edition is now available: click for details


Read All About It!

Monday, October 30, 2017 1:05 pm PST

My new book PQR Theory: How One Can Create a Universe is now available in print! Click here for details.

Be among the first to read about PQR Theory!

Thursday, March 17, 2016 6:18 pm PST

So far I’ve sent out about 60 preview copies of PQR Theory: How One Can Create a Universe.  It’s not too late to request yours–just click here to send an email to me at  No cost or obligation, but I’d love to hear what you think of it.

Have I solved the mystery of the Universe?

Monday, March 14, 2016 12:35 am PST

Pi Day (3.14.16) is here and my new book “PQR Theory: How One Can Create a Universe” is now ready for review. Once it appears in print, it will be bound to spark much discussion (and to hold the pages together), but you can be among the first to read it right now..

The book explains how a physical universe like ours can be created using numbers, and points the way to answering some age-old questions like:
–How can we have free will in a deterministic universe?
–What is time?
–What is gravity?
–Why is the space-time continuum an impossibility?
–What did Schrödinger’s cat eat?

Just click here to email me and I’ll send you a pdf of the advance edition for your review and comments. Limited time offer; no cost or obligation.

How One Can Create a Universe

Wednesday, February 17, 2016 8:52 am PST

Exciting news! Yesterday I finished the draft of my new book PQR Theory: How One Can Create a Universe. If you are interested in seeing an advance copy, please leave a comment, or you can email me privately using the link on

Soddy Quartets–an update

Sunday, February 16, 2014 12:13 pm PST

Some time ago I mentioned the apparent link between irreducible Soddy Quartets and the Ford Spheres that make up the 2-dimensional Flatrack Model. To recap, a Soddy Quartet is a set of four positive integers (a,b,c,d) that satisfy the equation

      (a+b+c+d)² = 3(a²+b²+c²+d²)

and it is said to be irreducible if these four numbers have no common factor greater than 1. As I noted previously, each such solution to this equation corresponds to exactly one group of four touching bubbles in the Flatrack model. The lines joining the largest three bubbles in such a group–those of orders a, b and c–define a triangle, within which the fourth bubble appears at a later stage (“time d”) in the model’s evolution. (For simplicity I have assumed that a, b, c and d are arranged in increasing order, since the model has a 6-way symmetry that corresponds to rearrangements of a, b and c.) Thus there is a one-to-one correspondence between irreducible Soddy Quartets and the triangles that are formed in the Flatrack model.

I’ve now taken this correspondence a step further. It turns out that any two Soddy Quartets that happen to share the same values of a, b and d, but have different c-values, correspond to two adjacent triangles in the Flatrack that together generate a Ford Sphere of order d. (And, not surprisingly, vice versa.)

For example the two quartets (4,19,25,39) and (4,19,37,39) each satisfy the above equation, and correspond to two adjacent triangles in the Flatrack: one defined by three touching bubbles of orders 4, 19 and 25; the other defined by the same bubbles of orders 4 and 19 plus another touching bubble of order 37. In other words, these two quartets correspond to two triangles that share a side in the Flatrack model. At “time” 39 this side is replaced by a new bubble of order 39 that touches all four of the preceding ones, and the two triangles that together generated this bubble are replaced by four new ones. (These four triangles correspond to the four Soddy Quartets (4,25,39,49), (4,37,39,61), (19,25,39,79) and (19,37,39,91), which can be considered as “existing” from “time” 39 until “times” 49, 61, 79 and 91 respectively.) And so the model evolves.

It seems that two Soddy Quartets can be considered “adjacent” in some sense if they have three numbers in common. I’m hoping that this can give rise to a meaningful measure of distance in the “space” formed by the solutions of this equation.

Now of course the next step is to extend this correspondence to the 3-dimensional model, so I’ve started investigating Gosset Quintets which are solutions to the equation

      (a+b+c+d+e)² = 4(a²+b²+c²+d²+e²)

and can be considered as corresponding to tetrahedrons that “exist” between “times” d and e in the Four-D Froth. However, these tetrahedrons appear to overlap or leave gaps in some places, so some refinement to the equations is needed here. Yet curiously enough as time increases, the total volume of these tetrahedrons consistently converges to just over 98% of the total volume in the model. I don’t know what the missing 2% represents, but I think I’m on a track worth pursuing.

A mechanism for gravity?

Thursday, July 26, 2012 1:11 pm PST

I recently realized something about the way my models could exhibit gravity-like phenomena.  If you recall, the models give a “space” that is constantly subdividing into ever-smaller granules as the “time” evolves.  (We then multiply by a scaling factor to keep the size of the newest granules constant, which produces an expanding model universe.)  The physical model can also be viewed as a “space” defined by an expanding graph of connected points, with new points continually appearing in between the existing ones.

If, however, this “space” does not subdivide where it is occupied by “matter” (or, more generally, if the regions of “space” that contain the most “matter” are the regions which subdivide the most slowly), then, when we apply the scaling factor, space would appear to expand fastest in those regions that contain the least matter.  Equivalently, the regions that contain matter would appear to be “pulling” on the rest of space, in proportion to the amount of matter they contain.

This could help explain why gravity is so weak in relation to the other fundamental forces of physics–it’s because the Universe is so old and has expanded so much during its long existence, that the gravitational effect is now greatly diluted in comparison to what it was in the Universe’s early stages, whereas the shorter-range forces are not.

Now since the subdivision of space and the passage of time are one and the same thing in my model, this means that when we measure time passing in a particular location, we are also monitoring the subdividing of space there.  So if I’m right in thinking that “occupied” space tends not to subdivide as much as “unoccupied” space, this would also imply that clocks run slower in regions that contain more matter–which is of course what we observe in our Universe, and what General Relativity describes.

To put it in a nutshell, “Matter eats Space and slows Time.”  And although I haven’t yet specified how “matter” occurs in my model, I’m thinking that maybe I should be looking for it where there’s the least amount of subdivision going on at any given “time.”  If I find it there, that would help explain how small-scale “quantum” processes can produce large-scale “relativistic” effects.

(Note: no apples were harmed in the conception of this blog post.)