J.Konstapel,Leiden,4-10-2016.
Used Blog: Exploring the Wheels and Mirrors of the Universe: From E8 to 8tonions
1. Introduction
This essay brings three structures together. Each comes from its own field. Each is well documented on its own. The claim of this essay is narrower than a theory of everything. The claim is this: at one specific place, the three structures produce the same numbers.
The three structures are:
- E8. This is the largest of the five exceptional Lie algebras. It has 248 dimensions and 240 root vectors. The physicist Frank (Tony) Smith has built a model of physics on it.
- The Bronze Mean. This is the number $\sigma = (3 + \sqrt{13})/2 \approx 3.3028$. It generates the sequence 1, 1, 4, 13, 43, 142.
- The Vacuum.Net theory. This theory writes every place in a net as a string of three-valued signs. Such a string is read as a number in balanced ternary.
The meeting point is the fifth ring of the net. A string of five signs has $3^5 = 243$ forms. Three of these forms are uniform: all plus, all rest, all minus. The other 240 match, in count, the 240 root vectors of E8.
Around this meeting point sit several further agreements. The gap between the binary ladder ($2^8 = 256$) and the ternary ladder ($3^5 = 243$) is 13. That number appears both in the Bronze Mean, under its square root, and in its sequence. The bronze sequence follows the rim of the rings for three steps. At its fourth step, the number 43, it leaves the rim and enters the fifth ring.
The method of this essay is deliberately modest. Every count is written out. Every count can be checked by hand. The final section keeps three categories apart: what is counted, what is read into the counts, and what remains open.
2. The net and its rings
2.1 Trits
The Vacuum.Net starts from a simple image: a fishnet in water. One strand crosses itself. At each crossing it does one of three things. It passes over. It passes under. Or it does not cross at all.
These are three states. Call them $+1$, $-1$ and $0$. One such sign is called a trit — the three-valued counterpart of the binary bit.
2.2 Strings and numbers
A place in the net is a string of trits. The string is read as a number in balanced ternary. The leftmost sign carries the highest power of three. A worked example from the source text:
$$+ – – – + = 81 – 27 – 9 – 3 + 1 = 43.$$
Three properties follow directly:
- Every whole number has exactly one such string.
- Zero is the string of rests.
- The mirror of a string — every sign reversed — is the negative of its number.
2.3 Rings
Strings of $n$ signs form ring $n$. Ring $n$ holds $3^n$ strings. The largest number in a ring is the string of $n$ plus signs. Call it the ceiling of the ring. The ceiling equals $(3^n – 1)/2$.
| Ring $n$ | Strings $3^n$ | Ceiling $(3^n-1)/2$ | Ceiling as string |
|---|---|---|---|
| 1 | 3 | 1 | + |
| 2 | 9 | 4 | ++ |
| 3 | 27 | 13 | +++ |
| 4 | 81 | 40 | ++++ |
| 5 | 243 | 121 | + |
| 6 | 729 | 364 | ++ |
The ceilings obey one simple rule: $c(n) = 3 \cdot c(n-1) + 1$. Each ring also holds exactly three uniform strings: all plus (the ceiling), all rest (zero), and all minus (the mirror of the ceiling).
2.4 The closure rule
The net has one rule for neighbours. Take a window of three neighbouring trits. The window is closed when the three signs add to zero, counted modulo 3. With two of the three signs fixed, exactly one value closes the window.
This rule matters later. It is the structural link to the concept of triality in E8.
3. The E8 model in brief
3.1 The construction
Smith builds his model on the Clifford algebra $\mathrm{Cl}(8)$. This algebra has $2^8 = 256$ dimensions, graded as:
$$1 + 8 + 28 + 56 + 70 + 56 + 28 + 8 + 1 = 256.$$
Two copies multiply to $\mathrm{Cl}(16) = \mathrm{Cl}(8) \otimes \mathrm{Cl}(8)$. Inside it lies E8, with 248 dimensions:
$$248 = 120 + 128.$$
The 120 is the algebra Spin(16). The 128 is one of its half-spinors. E8 has rank 8 and exactly 240 root vectors. These come in 120 pairs. Each root stands with its opposite.
3.2 The physical reading
Smith reads the 240 roots as physics:
- 24 + 24 roots from two copies of D4. One copy gives gravity. The other gives the forces of the Standard Model.
- 64 roots for eight dimensions of spacetime times eight Dirac gammas.
- 64 + 64 roots for eight kinds of fermion particles and eight kinds of antiparticles, each times eight.
The sum closes: $24 + 24 + 64 + 64 + 64 = 240$.
3.3 Three features of the model
Triality. The three blocks of 64 — spacetime, particles, antiparticles — are related by the triality of Spin(8). Triality binds three things of equal size so that two of them fix the third.
Three generations. A fermion of the first generation is one octonion. A fermion of the second is a pair. A fermion of the third is a triple. The generations are depths 1, 2 and 3.
The number 256. Smith’s book opens with $16 \times 16 = 256$ patterns and with the wheel of Ramon Llull: 16 points joined by 120 lines. Those 120 lines are the 120 generators of Spin(16).
4. Where two meets three
4.1 Two ladders
Smith counts in twos. Eight binary choices give $2^8 = 256$. The net counts in threes. Five trits give $3^5 = 243$.
The two ladders never coincide. The reason is exact: $\log_2 3 = 1.58496\ldots$ is not a fraction. But they come close where $3^q \approx 2^p$. Those places follow from the continued fraction of $\log_2 3$. Its convergents are 2/1, 3/2, 8/5, 19/12, 65/41 and further. The third convergent says: $3^5 \approx 2^8$.
4.2 The sandwich
| Number | What it is |
|---|---|
| 243 | $3^5$, the strings of ring 5 |
| 248 | the dimension of E8 |
| 256 | $2^8$, the dimension of Cl(8) |
E8 lies between the two ladders. It sits 5 above the ladder of three and 8 below the ladder of two. Those offsets are the exponents themselves. The whole gap is:
$$256 – 243 = 13.$$
The number 13 will return. It is the signature of the Bronze Mean.
5. The count
5.1 Removing the uniform rim
Take ring $n$ and remove its three uniform strings. What remains is $3^n – 3$.
| Ring $n$ | $3^n – 3$ | In the E8 model |
|---|---|---|
| 3 | 24 | the 24 roots of D4 |
| 5 | 240 | the 240 roots of E8 |
Both numbers are central in Smith’s structure. D4 appears twice inside E8. Its 24 roots form the figure called the 24-cell.
5.2 The classes agree
The grouping agrees as well. Rotate a string: move its first sign to the end. A string of three signs has three rotations. The 24 strings of ring 3 fall into 8 classes of 3. The 24-cell, in its standard description, consists of three figures of 8 points each. Triality carries one figure into the next. Both times the count is $3 \times 8$.
On ring 5, the 240 strings fall into 48 classes of 5. The number 48 is the number of roots of F4. F4 is the symmetry group of the exceptional Jordan algebra of 27 dimensions. That algebra has $3^3 = 27$ dimensions — the full count of ring 3. Smith also cites the ternary Golay code, with $3^6 = 27 \times 27 = 729$ words. That is the full count of ring 6.
5.3 The mirror agrees
The 240 roots form 120 pairs of opposites. Ring 5 holds the numbers 1 to 120 and their mirrors, −1 to −120. Zero and ±121 are the three uniform strings. So the positive roots match the numbers 1 to 120 in count. The mirror of a string matches the opposite of a root. In Smith’s reading, the opposite of a particle root is its antiparticle.
5.4 The inner shape of ring 5
One more count gives the inner shape of ring 5. Sort the 240 strings by their number of rests (zeros):
| Rests in the string | Strings |
|---|---|
| 0 | 30 |
| 1 | 80 |
| 2 | 80 |
| 3 | 40 |
| 4 | 10 |
The sum is 240. The first row is $2^5 – 2 = 30$: the strings without a rest, less the two uniform ones.
This table will matter in the final section. It shows that the 240 strings are not all alike. The 240 roots of E8, by contrast, are all alike.
6. Triality is the closure rule
Triality binds three things of equal size so that any two fix the third. The closure rule of the net says the same thing about three neighbouring trits. With two fixed, exactly one value closes the window.
Read as a trit, Smith’s three blocks are:
| Block | Trit value |
|---|---|
| Particle | +1 |
| Spacetime | 0 |
| Antiparticle | −1 |
Their sum is zero. A window holding all three values is one of the six closed windows that are not uniform.
The three generations fit the same ladder. One octonion, a pair, a triple: that is depth 1, 2 and 3 on a base of eight instead of three. Smith also finds a three-state system for the heaviest quark and the Higgs: a low, a middle and a high state, joined by one line. That, too, has the form of one trit.
7. The Bronze Mean
7.1 Definition
The Bronze Mean is the positive root of $x^2 – 3x – 1 = 0$:
$$\sigma = \frac{3 + \sqrt{13}}{2} = 3.30278\ldots$$
Its continued fraction consists of threes only: $3 + 1/(3 + 1/(3 + \ldots))$. Its sequence obeys the recurrence $b(n) = 3 \cdot b(n-1) + b(n-2)$:
$$1,\ 1,\ 4,\ 13,\ 43,\ 142,\ 469.$$
The steps are worth writing out: $4 = 3 \cdot 1 + 1$, $13 = 3 \cdot 4 + 1$, $43 = 3 \cdot 13 + 4$, $142 = 3 \cdot 43 + 13$. Each term is three times the one before, plus the one before that. The second part is the memory term. The sequence carries what lies two steps back.
7.2 The gap is thirteen
The gap between the ladders is $2^8 – 3^5 = 13$. Thirteen is a term of the bronze sequence. It is also the number under the root in $\sigma$.
7.3 The sequence leaves the rim at ring 5
Write each bronze term as a string:
| Bronze term | String | Ring | Ceiling of that ring |
|---|---|---|---|
| 1 | + | 1 | 1 |
| 4 | ++ | 2 | 4 |
| 13 | +++ | 3 | 13 |
| 43 | + − − − + | 5 | 121 |
| 142 | + − − + − + | 6 | 364 |
The check for 142: $243 – 81 – 27 + 9 – 3 + 1 = 142$.
The pattern is clear. The first three terms are the ceilings themselves. They are uniform strings. While the memory term is 1, the bronze rule and the ceiling rule are the same rule: three times the last, plus one.
At the next step the two rules part. The ceiling of ring 4 is $3 \cdot 13 + 1 = 40$. The bronze term is $3 \cdot 13 + 4 = 43$. The number 43 is larger than 40. It does not fit on ring 4. It moves to ring 5, where it is the string $+ – – – +$.
So the memory term does one thing. It lifts the sequence off the uniform rim and into the interior of ring 5. That interior is the set of 240 strings counted in Section 5. The number 43 is the first bronze term that is not a ceiling. In the count, it is one of the 240.
The difference is $43 – 40 = 3$. And the 240 itself is $243 – 3$.
7.4 A place for an old pair of numbers
This gives a place to an old pair of numbers. Built labyrinths count to forty: the classical one counts $7 \cdot 5 + 5 = 40$, the one in Chartres $11 \cdot 3 + 7 = 40$. The Sri Yantra has 43 triangles. Forty is the rim of ring 4. Forty-three is the first step inside ring 5.
7.5 The fourth power is 119
The powers of $\sigma$, taken with their mirror, give whole numbers: $\sigma^n + (-1/\sigma)^n$. They obey the same recurrence, starting from 2 and 3:
$$2,\ 3,\ 11,\ 36,\ 119,\ 393.$$
So $\sigma^4 + \sigma^{-4} = 119$, exactly. The roots of E8 form 120 pairs. The two numbers differ by one.
The same 119 occurs elsewhere. From the Planck length to the width of the observable universe is a factor of about $5 \times 10^{60}$. Since $\log_{10} \sigma = 0.5189$, that is 119 bronze steps: $119 \times 0.5189 = 61.7$.
8. Gold and bronze
In Smith’s book the mean is gold, not bronze. He builds E8 from two copies of a figure with 120 points, related by the Golden Ratio. That gives $120 + 120 = 240$.
Gold and bronze are family. Both are metallic means: the roots of $x^2 – m \cdot x – 1 = 0$.
| Mean $m$ | Value | Continued fraction | Symmetry of its tiling |
|---|---|---|---|
| Gold 1 | 1.618… | all ones | fivefold |
| Silver 2 | 2.414… | all twos | eightfold |
| Bronze 3 | 3.303… | all threes | sixfold |
Gold belongs to the division in five. Bronze belongs to the division in three. E8 carries both — and the count of Section 5 shows how. The 240 roots are the strings of five signs on a base of three. The five is the side that Smith describes. The three is the side of the net.
9. What this says about the MAZE
The MAZE is the address system built on the net. Every item has a number. The number is a string of trits. Five conclusions follow:
Ring 5 is the ring of E8. The MAZE has 243 addresses on ring 5. Take away the rest string and the two ceilings, and 240 remain.
The ceilings are the uniform strings. The series 1, 4, 13, 40, 121 is the rim. The roots are everything between.
The mirror is the antiparticle. The mirror column of the MAZE — $n$ against $-n$ — is the same operation as root against opposite root.
The 256 and the 243 are neighbours. A byte holds 256 values. An address of five trits holds 243. Ring 5 is the place where the two-valued world of present computers and the three-valued MAZE can be exchanged, with a remainder of 13.
Smith supplies inhabitants. The MAZE has the addresses of ring 5. Smith has names for 240 roots: forces, spacetime, particles, antiparticles. Joined, that is a body of physics with an address by construction.
10. Status: counted, read, open
This essay keeps three categories strictly apart.
10.1 Counted — exact and checked
- $3^5 – 3 = 240$ and $3^3 – 3 = 24$.
- 8 classes of 3 on ring 3; 48 classes of 5 on ring 5.
- 120 mirror pairs on ring 5.
- $2^8 – 3^5 = 13$, and $243 < 248 < 256$.
- The bronze terms 1, 4, 13 are ceilings; $43 = + – – – +$ lies on ring 5; $142 = + – – + – +$ lies on ring 6.
- $\sigma^4 + \sigma^{-4} = 119$.
10.2 Read — interpretations, not derivations
- Triality as the closure rule, with particle, spacetime and antiparticle as +1, 0 and −1.
- The three generations as depths 1, 2 and 3.
- The memory term as the step from the rim into ring 5.
10.3 Open — two things not yet there
1. The map. The counts agree, but no rule yet says which root belongs to which string. There is a real obstacle. All 240 roots of E8 are alike: the symmetry of E8 carries any root to any other. The 240 strings are not alike. They differ in their number of rests, as the table in Section 5.4 shows. A map must say what those five kinds become on the side of E8.
2. The sum. In E8, two roots either add to a third root or they do not. Each root has exactly 56 partners with which it adds to a root. In the MAZE, two addresses add with a carry. If the map exists, the two sums can be compared: 240 roots against 240 strings, one sum on each side. If the sums agree, the open rewrite rule of the MAZE is the structure of E8. If they do not, the agreement is a count and no more.
The Bronze Mean keeps its earlier status. It is not derived from closure here. What this essay adds is a place: the point where the bronze sequence and the rim of the rings part is the ring where E8 lives.
11. Conclusion
Three structures, three fields, one meeting point. The fifth ring of the Vacuum.Net holds 243 strings. Remove the uniform rim of three, and 240 remain — the exact number of the root vectors of E8. The gap between the binary and ternary ladders at this point is 13, the signature number of the Bronze Mean. The bronze sequence itself runs along the rim for three steps, then steps into the interior of ring 5 at the number 43.
None of this yet constitutes a derivation. The counts are exact. The readings are suggestive. The map remains open. But the pile of agreements — 240, 120, 48, 24, 13, 119 — is dense enough at one single place to justify the next step: writing down the map, and comparing the sums.
Annotated References
*Smith, F. D. (Tony), Jr. (2008). Physics of E8 and Cl(16) = Cl(8) ⊗ Cl(8). Georgia, June–July 2008.*
Why read it? This is the source of the E8 model used throughout the essay: the 256 dimensions of Cl(8), the split 248 = 120 + 128, the physical reading of the 240 roots, triality among the three blocks of 64, the three generations, and the construction of E8 from two golden figures. How to read it: start with the Technical Introduction and “Structure of the E8 Physics Model”; then move to “H4 + H4 = E8′”. This is the primary technical reference for Sections 3 and 8.
Konstapel, J. (2026). The Climate on the Ladder of Three. Constable Research, Leiden, 3 October 2026.
Why read it? Derives the layers from the trit, states the closure rule on three neighbours, and locates the closure depths where $3^q \approx 2^p$. How to read it: Section 3 is the relevant part. Grounds Sections 2 and 4 of this essay.
Konstapel, J. (2026). Everything Has Its Number in the Maze. Constable Research, Leiden, August 2026.
Why read it? Establishes the unique address in balanced ternary. Works out $43 = + – – – +$ as its central example. Covers the mirror and the series of ceilings. How to read it: use as the companion to Sections 2 and 9.
Konstapel, J. (2026). Counting the Universe. Constable Research, Leiden, August 2026.
Why read it? Introduces the bronze sequence and the memory term. Tests 43 against the ceiling 40. Computes the 119 bronze steps from the Planck length to the observable universe. How to read it: the background for Section 7.
Konstapel, J. (2026). The Labyrinth Counts to Forty. Constable Research, Leiden, September 2026.
Why read it? Explains why built labyrinths count 40 and not 43: they record the rim, not the rule of growth. How to read it: short piece; supports Section 7.4.
Baez, J. C. (2002). “The Octonions.” Bulletin of the American Mathematical Society 39, 145–205.
Why read it? The clearest standard account of triality, the exceptional Jordan algebra of 27 dimensions, F4 and E8, written for mathematically literate readers. How to read it: focus on the parts on triality and on the exceptional Jordan algebra. Independent mathematical backing for Sections 3 and 5.
Conway, J. H., and Sloane, N. J. A. (1999). Sphere Packings, Lattices and Groups, third edition. New York: Springer.
Why read it? The reference work on the 240 root vectors of E8 and their inner products. Also covers the binary and ternary Golay codes. How to read it: Chapter 4 is the relevant chapter. Supports the counts in Section 5.
Coxeter, H. S. M. (1973). Regular Polytopes, third edition. New York: Dover.
Why read it? Describes the 24-cell and its three inscribed figures of eight points. Also treats the figure of 120 points behind the golden construction of E8. How to read it: consult the chapters on four-dimensional polytopes. Supports Sections 3 and 8.
Moody, R. V., and Patera, J. (1993). “Quasicrystals and icosians.” Journal of Physics A 26, 2829–2853.
Why read it? Shows how E8 falls into two golden copies of a figure with 120 points. This is the mathematics behind Smith’s “H4 + H4 = E8′”. How to read it: read after Coxeter; directly relevant to Section 8.
de Spinadel, V. W. (1999). “The family of metallic means.” Visual Mathematics 1(3).
Why read it? Presents gold, silver and bronze as one family, with their continued fractions. How to read it: introductory; the frame for Section 8.
Dotera, T., Bekku, S., and Ziherl, P. (2017). “Bronze-mean hexagonal quasicrystal.” Nature Materials 16, 987–992.
Why read it? Realises the Bronze Mean as a sixfold tiling of triangles and rectangles. Shows that bronze is the mean of the division in three. How to read it: a physics-periodical confirmation that the bronze mean is not a numerical curiosity.
Hardy, G. H., and Wright, E. M. (1938). An Introduction to the Theory of Numbers. Oxford: Clarendon Press.
Why read it? The classical treatment of continued fractions and their convergents — the ground under $3^5 \approx 2^8$ and under $\sigma$ as a continued fraction of threes. How to read it: Chapter X.
Knuth, D. E. (1997). The Art of Computer Programming, Vol. 2, §4.1. Reading: Addison-Wesley.
Why read it? The standard reference on balanced ternary: every whole number has exactly one string, and the mirror of a string is its negative. How to read it: §4.1 is short and self-contained. Supports Section 2.