have a look at “About Saturn, the Son of the Sun “
J.Konstapel,2-10-2026.
From Quaternions to Vacuum.Net: Mapping Four Theories onto Cl(3)
skip the dutch intro here
Sommige theorieën over de natuur lijken elkaars tegenpolers: de klassieke natuurkunde van Maxwell, de kosmologie van Walter Russell, het bewegingssysteem van Dewey Larson en de hedendaagse Vacuum.Net-theorie. Een essay op constable.blog betoogt dat deze vier op dezelfde wiskunde rusten. Die wiskunde heet de Clifford-algebra van de ruimte, afgekort Cl(3). Wat volgt legt uit wat dat is, en waarom het voor alle vier werkt.
Stap 1: de bouwstenen van de ruimte
Stel je een lege ruimte voor met drie richtingen: voor-achter, links-rechts, boven-onder. De Clifford-algebra stelt bij die drie richtingen één vraag: welke combinaties kun je maken? Het antwoord is een kleine, complete lijst:
- één punt van stilte (de “scalar” — een getal zonder richting);
- drie lijnen (de drie assen);
- drie vlakken (elk gevormd door twee assen);
- één volume (de drie assen samen).
Tel ze op: 1 + 3 + 3 + 1 = 8 bouwstenen. Meer heb je voor een beschrijving van de driedimensionale ruimte niet nodig: alles is een punt, een lijn, een vlak of een volume — of een som daarvan. Dat is de kern van Cl(3).
Stap 2: de kubus als telraam
Hoe onthoud je die acht bouwstenen met hun richtingen? Neem een kubus en geef elke as één teller met drie standen: 0 (de as doet niets mee), + (de as doet mee, positief gericht) of − (de as doet mee, negatief gericht). Elke combinatie van drie tellers is één cel van de kubus. Omdat elke as drie standen heeft, ontstaan er 3 × 3 × 3 = 27 cellen.
Die 27 cellen bevatten precies de acht bouwstenen van stap 1, nu mét richting: één cel met drie nullen is het middelpunt (het punt van stilte), zes cellen met twee nullen liggen op de assen (lijnen), twaalf cellen met één nul liggen in de vlakken, en acht cellen zonder nul vullen de hoeken (volumes). De kubus is dus een telraam waarin de hele algebra zit.
Stap 3: de selectieregel — wat blijft bestaan
Van de 27 cellen blijven er volgens één regel maar negen over: een cel blijft bestaan wanneer haar drie tellers elkaar opheffen, dus wanneer plus en min in balans zijn. Die negen vormen twee herkenbare patronen:
- een platte schijf van zes cellen rond het middelpunt. Wiskundig gezien is dit de quaternionen-algebra: het standaardgereedschap voor draaiingen. Elke draaiing in de ruimte is hiermee te schrijven, en elke draaiing hoort bij twee schijfcellen — één toestand komt altijd dubbel voor.
- één as van hoek tot hoek, met aan elk uiteinde één cel: + + + + + en − − − − −. Deze as geeft de draairichting aan: linksom of rechtsom, zoals je spiegelbeeld.
De kern van het essay: dit ene beeld — schijf plus as, negen cellen in een kubus van 27 — is de gemeenschappelijke structuur van vier theorieën.
Stap 4: de vier theorieën op het telraam
Maxwell (1873). Maxwell schreef zijn veldwetten in quaternionen. Elke wiskundige bewerking op een veld leverde hem twee dingen tegelijk op: een grootheid zonder richting (die hij “convergentie” noemde) en een grootheid mét richting (de “rotatie”). Op het telraam is dat precies de middencel en de schijf: het punt van stilte en de draaiing. Omstreeks 1884 vereenvoudigden navolgers zijn rekenwerk tot losse “dot”- en “cross”-producten — handig om mee te rekenen, maar het middelpunt en het vlakkarakter verdwenen uit het beeld.
Walter Russell (1926). Russell tekende de natuur als een golf die vanuit een stil middelpunt spiraalt naar de hoeken van een kubus. Zijn bouwstenen — middelpunt, assen, vlakken, acht hoeken — liggen één op één op het telraam van stap 2. Zijn twee hoofdkrachten, generatie (samentrekken naar het midden) en radiatie (uitdijen vanaf het midden), vallen samen met de twee uiteinden van de as uit stap 3. En koolstof, de top van zijn toonladder, is de enige toon die volledig ín de schijf ligt: een zuivere, rustige draaiing. Daarmee verklaart het essay waarom koolstof bij Russell zo uitzonderlijk symmetrisch is.
Dewey Larson (1959). Larson classifyeerde alle bewegingen naar dimensie-aantal: elektrische verschijnselen zijn ééndimensionaal, magnetische tweedimensionaal, zwaartekracht driedimensionaal. Op het telraam is dat de opsomming van lijn, vlak en volume uit stap 1 — dimensie voor dimensie identiek.
De Vacuum.Net-theorie. Deze theorie beschrijft het vacuüm als één streng die zichzelf kruist. Elke kruising krijgt een van drie waarden: over, onder, of geen kruising — plus, min of nul, dus precies de drie standen van elke as op het telraam. En haar kernregel luidt: wat sluit, blijft bestaan. Dat is letterlijk de selectieregel van stap 3. De theorie levert daarmee de motor die het telraam laat draaien.
Wat het essay laat zien
Vier denkers werkten onafhankelijk van elkaar, in verschillende tijden en talen. Het essay toont dat zij dezelfde acht bouwstenen gebruikten, dezelfde drie dimensies telden en dezelfde scheiding maakken tussen rust in het midden en richting op de as. Maxwell rekende het uit, Russell tekende het, Larson classificeerde het, en de Vacuum.Net-theorie vat het samen in één regel. Verschillende dialecten, één grammatica.
Bron: From Quaternions to Vacuum.Net: Mapping Four Theories onto Cl(3) — constable.blog, 2 oktober 2026.
J.Konstapel,Leiden,2-10-2026.
Intro
Four very different bodies of work describe the physical world.
1 Clerk Maxwell wrote electromagnetism in quaternions in 1873.
2 Walter Russell drew a cosmos of two opposed spirals in a cube of stillness between 1926 and 1953.
3 Dewey Larson built a universe of pure motion in 1959.
4 The Vacuum.Net theory describes the vacuum as one strand that crosses itself, counted in three values.
This essay shows that all four rest on one and the same mathematical structure: Cl(3), the Clifford algebra of three-dimensional space.
The key is a simple identity. Give each axis of a cube one “trit” — absent (0), or present with a sign (+1 or −1). The resulting 27 cells are exactly the signed building blocks of Cl(3), because (1 + 2)³ = 27.
2. Why this matters
Each of the four works was built in isolation.
Maxwell had no reason to read Russell. Russell rejected Maxwell’s successors.
Larson built his system against both relativity and quantum theory.
The Vacuum.Net theory began from a single strand and a count.
Yet each describes the same few things: a still reference, two opposed directions, lines, planes and volumes, and a rule that decides which configurations persist. The question is whether this is resemblance or identity. Resemblance allows a list of parallels. Identity requires one structure on which all four can be written without remainder. This essay claims identity, and tests each work against that claim.
3. The Clifford algebra of space, briefly
William Kingdon Clifford defined his algebra in 1878. It unites Hamilton’s quaternions with Grassmann’s exterior algebra. For three-dimensional space it is written Cl(3), and its rules are few.
- There are three basis vectors: e₁, e₂ and e₃.
- Each squares to +1.
- Any two different ones anticommute: e₁e₂ = −e₂e₁.
Products of basis vectors are called blades. The number of factors is the blade’s grade.
| Grade | Elements | Count | Geometric meaning |
|---|---|---|---|
| 0 | 1 | 1 | a scalar |
| 1 | e₁, e₂, e₃ | 3 | oriented lines |
| 2 | e₁₂, e₂₃, e₁₃ | 3 | oriented planes |
| 3 | e₁₂₃ = I | 1 | an oriented volume |
That gives 1 + 3 + 3 + 1 = 8 basis blades. Four facts carry the rest of the essay.
- Bivectors square to −1. The unit bivectors behave like the imaginary unit i, j and k.
- The even part is the quaternions. The scalar plus the three bivectors closes under multiplication. This is Hamilton’s quaternion algebra. Every rotation is described by two quaternions, q and −q. That is the double cover.
- The pseudoscalar is handedness. I squares to −1 and commutes with everything. Its sign distinguishes a right-handed from a left-handed frame. A mirror reflection flips it.
- Duality joins planes and lines. Multiplying by I turns a vector into the bivector perpendicular to it. The product of two vectors splits into a scalar part (the dot product) and a bivector part (the outer product).
4. The trit cube: the skeleton of the algebra
Take a cube and its three axes. On each axis place one trit with this reading:
- 0 — the axis does not take part;
- +1 or −1 — the axis takes part, with that sign.
Each cell (x, y, z) of the cube then defines one blade, and its grade is the number of axes that do not read 0. The count per grade is striking:
| Grade | Choice of axes | Signs | Cells |
|---|---|---|---|
| 0 | 1 way | 1 | 1 |
| 1 | 3 ways | 2 | 6 |
| 2 | 3 ways | 4 | 12 |
| 3 | 1 way | 8 | 8 |
The total is 1 + 6 + 12 + 8 = 27, which is (1 + 2)³ = 3³. The 1 stands for an absent axis; the 2 stands for a present axis with its sign. The trit cube contains every blade of Cl(3), each with every factorisation of its sign.
The grades also have a geometric place in the cube. The cell with three zeros is the centre point. The six cells with two zeros lie on the axes. The twelve cells with one zero lie in the median planes. The eight cells without a zero fill the eight corners. Point, lines, planes and volumes: the algebra’s grades and the cube’s geometry are the same partition.
One caveat. Cl(3) carries real coefficients — magnitudes. The trit cube carries none. It records which blade and which sign, but not how much. It is the combinatorial skeleton of the algebra; the magnitudes live in a separate register of tension.
5. Closure: the nine cells that persist
In the Vacuum.Net theory, three trits close when their sum is 0, counted modulo 3. Applied to the cube: a cell closes when x + y + z ≡ 0 (mod 3). Of the 27 cells, exactly 9 close.
- The centre cell (0, 0, 0) is the scalar 1.
- Six cells form a regular hexagon around the centre, in the plane x + y + z = 0. They carry the three unit bivectors, each twice.
- The two remaining cells, (1, 1, 1) and (−1, −1, −1), sit at the ends of the main diagonal. They carry +I and −I.
Three consequences follow.
The disc is the quaternion frame. The centre and the hexagon carry the scalar and the three bivectors — the four quaternion units, up to sign. The closing disc is the even subalgebra of Cl(3).
The double cover becomes visible. Each unit bivector comes from two cells with opposite factor signs: (1, −1, 0) and (−1, 1, 0) both give −e₁₂. Two cells, one rotation.
The axis is handedness. The disc’s axis is the pseudoscalar, and its two ends are the two handednesses. Closures also compose: adding two closing cells modulo 3 yields another closing cell. The nine cells form a group.
6. The four works on this one structure
6.1 Maxwell (1873)
Maxwell wrote his Treatise on Electricity and Magnetism with quaternions alongside Cartesian components. The operator ∇ acted on a vector field and returned a full quaternion. Its scalar part he called the convergence (the divergence with a minus sign). Its vector part he called the curl. On Cl(3), this is simply the split of a product into grade 0 and grade 2. Maxwell’s quaternion calculus is the calculus of Cl(3), seen through its even subalgebra.
The field itself splits by grade. In modern geometric algebra, the electromagnetic field is one object: F = E + IB. The electric field is a vector (grade 1). The magnetic field enters as a bivector (grade 2) — an oriented plane, not an arrow. That is why it behaves as a pseudovector under reflection.
Maxwell also kept a scalar slot in every product. His Article 617 sets the convergence of the vector potential to zero. In the trit cube that slot is the centre cell: the same place where Russell puts his still centre and Larson his scalar progression.
6.2 The amputation of 1884–85
Oliver Heaviside and Josiah Willard Gibbs replaced the single quaternion product by two separate products: the dot product and the cross product. The equations became shorter and easier to compute. But on Cl(3), two things were lost. The grade-0 slot was detached from the field. And the grade-2 nature of magnetism was hidden behind an axial vector — an arrow standing for a plane. In the trit cube, that is the removal of the centre of the closing disc, and the collapse of twelve plane cells onto six axis cells. Russell’s later complaint — that two poles cannot hold a three-dimensional universe, but four can — is the same loss, seen from his side.
Maxwell himself had left one grade open. In 1865 he attempted a field theory of gravitation. The energy of the gravitational field came out negative, and he stopped. On Cl(3), gravitation belongs to the pseudoscalar grade — the one grade his field equations do not contain.
6.3 Russell (1926–1953)
In The Universal One, every wave lives in a cube of motion. Three median planes divide it into eight compartments. The wave begins at the centre and spirals to the corners. On Cl(3) this is exactly the grade partition of the trit cube:
- Russell’s centre is the scalar — grade 0.
- His electric pole, an axis, is a vector — grade 1.
- His planes of inertia are bivectors — grade 2.
- His eight compartments are trivectors — grade 3.
Russell has two forces: generation, which closes toward a centre, and radiation, which opens away from it. He also fixes one sense of rotation for all motion. With the rotation fixed, the direction along the axis alone decides the handedness of the spiral. Generation and radiation are therefore +I and −I: the two poles of the closing axis, the only two closures without a zero.
His octave of tones runs from 0 through 1+, 2+, 3+ to the double tone 4±, and back to 0. Carbon sits at 4±, where the spiral is flattened into a watch spring. A flat spiral has no component along the axis. It lies entirely in the closing disc — the even subalgebra. Carbon is a pure quaternion rotation. That is why Russell finds at carbon what he finds nowhere else: one plane, no precession, true spheres and true cubes. Every other tone carries a residual component along the axis, which Russell measured as oblateness.
In 1953 Russell insisted that every orbit is controlled by four magnetic poles, not two. A vector has two ends; a quaternion has four components. His four poles are the quaternion frame of the closing disc.
One point does not fit cleanly. Russell divides his eight compartments into a positive and a negative half by one plane. Cl(3) divides its eight volumes by parity, giving two interpenetrating tetrahedra. The count agrees; the arrangement does not.
6.4 Larson (1959)
Larson’s Reciprocal System starts from one postulate: the universe consists of motion only. Space and time are the two reciprocal aspects of that motion. His reference is the scalar progression — the outward motion of the natural reference system at unit speed. Against it he classifies motions by dimension:
- one-dimensional motion is electric;
- two-dimensional motion is magnetic;
- three-dimensional motion is gravitation, inward.
These are the four grades of Cl(3), one for one. Larson’s opposition of inward gravitation and outward progression is a sign — the sign of the pseudoscalar, the same axis that carries Russell’s generation and radiation.
One difference is fundamental. Larson insists his motions are scalar: magnitude without direction in space. Cl(3)’s blades are oriented. The fit holds in the number of dimensions per motion; the orientation is what Cl(3) adds. In the trit cube, that orientation is the sign of each factor.
6.5 The Vacuum.Net theory
The Vacuum.Net theory describes the vacuum as one strand that crosses itself. At each crossing the strand runs over, runs under, or does not cross: +1, −1 or 0. On the trit cube, the strand’s configurations are the oriented cells of Cl(3), and the net’s rule — what closes, remains — selects the disc and the axis. A winding that persists is either a pure rotation in the even subalgebra, or pure handedness on the axis.
The structure also explains a repair the measurements had already required. The net’s composition rule R_Q combines neighbours as unit quaternions. Averaging quaternions directly can collapse, because q and −q describe the same rotation. The repair was a sign-invariant measure. In the trit cube this is not a technicality: each closing bivector occupies two cells, and a measure that respects the cube must count the pair as one.
7. The dictionary at a glance
| Grade | Cl(3) | Trit cube | Russell | Larson | Maxwell |
|---|---|---|---|---|---|
| 0 | scalar 1 | centre cell | still centre | scalar progression | scalar slot, convergence |
| 1 | vectors | 6 axis cells | electric pole | electric, 1D | electric field E |
| 2 | bivectors | 12 plane cells | planes of inertia | magnetic, 2D | magnetic field IB |
| 3 | trivector I | 8 volume cells | compartments | gravitation, 3D | gravitation, left open (1865) |
| even part | quaternions | closing disc | carbon, four poles | — | Treatise notation |
| double cover | q ≡ −q | two cells per bivector | — | — | — |
8. Consequences
Carbon. Carbon is the one tone of Russell’s octave that lies in the closing disc — a pure rotation with no residual component along the axis. At carbon both triangles of the hexagon are available, so every closure has its mirror beside it. A form that copies itself from carbon selects one triangle, and so one handedness. Living matter built on carbon has one handedness. Silicon holds the same position one octave deeper: the same pure rotation, at a greater depth.
Time and the scalar. Every theory here places something special at grade 0. Russell puts the still centre there. Larson puts the progression of space and time there. Maxwell put the scalar slot there, which carried charge. In William Baylis’s algebra of physical space, time is the scalar part of a paravector. The scalar grade is where the reference is kept. Removing it, as in 1884–85, removes the reference from the field.
Electric and magnetic as one and two. Larson’s claim that electricity is one-dimensional and magnetism two-dimensional is not, on this structure, a hypothesis. It is how Cl(3) writes Maxwell’s field. Russell’s later placing of magnetism in the planes of stillness is the same statement in his language.
Handedness as the poles of sameness. The two closures without a zero close by sameness: all three factors alike. The seven closures with a zero close by difference. On Cl(3), sameness is pure handedness; difference is rotation. The essay extends this to human systems: a workshop or a market holds people who differ and still fit — closure in the disc. A system of interchangeable users holds by sameness — closure at the poles.
9. What stays open
Three points do not yet fit without remainder.
- Larson’s motions are scalar in every dimension; Cl(3) orients them. The fit holds in the count of dimensions, not in direction.
- Russell’s division of his compartments into two halves differs from the parity division of Cl(3) into two tetrahedra.
- The trit cube carries signs, not magnitudes. The net keeps magnitudes in a separate register of tension. How the two registers compose on the full algebra is the next derivation.
Annotated references
Each entry notes why it is worth reading. Most are primary sources; a few are modern treatments or historical accounts.
- Clifford, W. K. (1878). “Applications of Grassmann’s Extensive Algebra.” American Journal of Mathematics 1(4), 350–358.
Why read? The founding paper of the algebra that unites Hamilton and Grassmann — the basis of the entire essay. - Hamilton, W. R. (1853). Lectures on Quaternions. Hodges and Smith, Dublin.
Why read? The quaternions in their inventor’s form, with the scalar and vector parts that Maxwell used. - Hestenes, D. (1966). Space-Time Algebra. Gordon and Breach, New York.
Why read? The book that brought Clifford algebra back into physics and wrote Maxwell’s equations as one equation. - Hestenes, D. (1986). New Foundations for Classical Mechanics. Reidel, Dordrecht.
Why read? Cl(3) as the language of three-dimensional mechanics: rotations by quaternions, the pseudoscalar, duality. Reading advice: the first two chapters suffice for this essay. - Doran, C. & Lasenby, A. (2003). Geometric Algebra for Physicists. Cambridge University Press.
Why read? The standard modern treatment, including the single-object field F = E + IB. - Baylis, W. E. (1999). Electrodynamics: A Modern Geometric Approach. Birkhäuser, Boston.
Why read? The algebra of physical space: time as the scalar part of a paravector. - Maxwell, J. C. (1865). “A Dynamical Theory of the Electromagnetic Field.” Philosophical Transactions of the Royal Society 155, 459–512.
Why read? Section 82 contains the attempt at a field theory of gravitation and the point where Maxwell stopped. - Maxwell, J. C. (1873). A Treatise on Electricity and Magnetism, 2 vols. Clarendon Press, Oxford.
Why read? The quaternion form of the field; convergence and curl as scalar and vector parts. Reading advice: Volume 1, Article 25 for ∇; Volume 2, Articles 591–619 for the general equations. - Heaviside, O. (1893). Electromagnetic Theory, Vol. 1. The Electrician, London.
Why read? The vector form that replaced Maxwell’s quaternions, and Heaviside’s own arguments for it. - Crowe, M. J. (1967). A History of Vector Analysis. University of Notre Dame Press.
Why read? The full history of the change from quaternions to vectors, 1844–1910. - Russell, W. (1926). The Universal One. Reprint 1974, University of Science and Philosophy, Waynesboro, with a preface by Lao Russell.
Why read? The cube of motion, the cone from hydrogen to carbon, and the octave formula 4± 3+ 2+ 1+ 0 1− 2− 3− 4±. - Russell, W. (1947). The Secret of Light. Walter Russell, New York.
Why read? The still magnetic Light and the two lights that never reach unity. - Russell, W. (1953). A New Concept of the Universe. Walter Russell Foundation, Swannanoa.
Why read? The wave field, and the claim of four magnetic poles. - Larson, D. B. (1959). The Structure of the Physical Universe. North Pacific Publishers, Portland.
Why read? The Reciprocal System: motion only, scalar progression, and motion in one, two and three dimensions. - Larson, D. B. (1979). Nothing but Motion. North Pacific Publishers, Portland.
Why read? The revised statement of the system’s foundations. - Rowlands, P. (2007). Zero to Infinity: The Foundations of Physics. World Scientific, Singapore.
Why read? A physics built on quaternions and vectors with a zero-sum condition — close to the structure presented here. - Crowell, R. H. & Fox, R. H. (1963). Introduction to Knot Theory. Ginn, Boston.
Why read? Crossing signs and tricolouring, to which the closure condition is equivalent. - Konstapel, J. (2026). The Cube, the Cone and Carbon. Constable Research, Leiden, 1 October 2026.
Why read? Russell’s theory and geometry, and their step-by-step translation into the net. - Konstapel, J. (2026). Clerk Maxwell and the Vacuum.Net Theory, sixth edition. Constable Research, Leiden.
Why read? Maxwell’s quaternion field rewritten as the net: the composition rule R_Q and the sign-invariant ring measure. - Konstapel, J. (2026). “Larson, Grassmann en de Vacuum.Net-theorie.” constable.blog, 27 September 2026.
Why read? Larson and Grassmann set beside the net. - Konstapel, J. (2026). “Why Vacuum.Net Is a Swarm.” constable.blog, 23 September 2026.
Why read? The count of 9 closing triples out of 27.