Chemistry explained by one simple fractal rule -1<0<1

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Het boek beweert dat de hele scheikunde volgt uit één simpele regel: drie waarden (−1, 0, +1) die samen nul zijn, waarbij wat opsluit blijft bestaan en wat niet sluit verder gaat als spanning (reactiviteit).

Uit die ene regel volgt het periodiek systeem: alle 118 elementen komen in hun juiste periode terecht, en het atoom blijkt niets meer dan een nette telling van “plaatsen” en “ringen”.

Een chemische binding is gewoon een ontmoeting van twee tegengestelde waarden die samen sluiten — en daarmee worden zelfs de lastige gevallen (magnetisch zuurstof, koolmonoxide) vanzelf verklaard.

De vormen van moleculen (tetraëder van koolstof, 109,47°) en links- versus rechtsdraaiendheid (chiraliteit) volgen uit de hoeken van een kubus die in de drie waarden verborgen zit.

Het boek telt alle denkbare moleculen: het aantal alkanen en alcoholen, de suikers, de twintig aminozuren en de twee DNA-basisparen A·T en G·C — telkens exact gelijk aan wat de scheikunde al wist.

Zelfs ijs volgt mee: de restentropie van ijs wordt tot op vijf decimalen exact berekend, gelijk aan een beroemd wiskundig resultaat van Lieb.

De grens is eerlijk toegegeven: echte energieën in elektronvolt kan de theorie nog niet voorspellen — maar de architectuur van de scheikunde blijkt te kunnen met rekenen op een papieren servet, zonder kwantummechanica of kwantumcomputer.

J.Konstapel,Leiden,11-11-2026.

Every science accumulates rules, and chemistry has more than its share of them. There is a rule for how electrons fill the shells of atoms (Madelung), a rule for how they line up their spins (Hund), a rule for which atoms stick together and how many bonds they form (Lewis), a rule for the shapes of molecules (VSEPR), a rule for how bond lengths shrink as bonds grow stronger (Pauling), a rule for which ring molecules are unusually stable (Hückel). Each of these was discovered separately, between 1916 and the 1950s, and each has been explained, since the advent of quantum mechanics in 1927, by solving the same equation in a different setting.

The Chemistry of the Vacuum.Net Theory is a book with a bet in it. The bet is this: all of those rules — and a good deal more — can be derived from a single, almost childishly simple starting point, without quantum mechanics, without wave functions, and without the enormous computers that simulate molecules today. The starting point is what the author calls the trinity: three values, −1, 0 and +1, whose sum is zero. Everything else — the periodic table, the magnetism of atoms, the shapes of molecules, the number of possible sugars, the two base pairs of DNA, even the residual entropy of ice — is claimed to follow from counting with those three values.

What makes the book unusual, and worth taking seriously, is its method. This is not a philosophical tract dressed in chemical clothing. Every claim is turned into a small computer program, every program produces numbers, and every number is compared with measured data. The book reports its hits and its misses side by side, using four honest labels: derived (computed before comparison), reproduces (re-derives a law chemistry already knew), fitted (a choice made after seeing data, and admitted as such), and open (not yet done). An independent check of the book’s calculations confirms that essentially everything re-computable matches exactly.

This essay explains what the theory says, how it conjures chemistry out of arithmetic, where it succeeds, and where it stops.

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Part I — The ground: three values and a carry

The theory begins with balanced ternary arithmetic, an old and elegant way of writing numbers using only −1, 0 and +1. Every whole number has exactly one such representation. The number 43, for example, is 81 − 27 − 9 − 3 + 1, written 1, −1, −1, −1, 1. Negation is simply flipping every digit — no minus sign needed.

The single dynamical rule is the carry, familiar from ordinary addition. When you add digits, the remainder stays where it is and the rest moves, whole and undivided, to the next scale. The book reads this dry fact as a physics:

• 1 + (−1) = 0: two opposite values meet and close. Nothing passes on. In chemistry, this is the chemical bond.
• 1 + 1 + 1 = 0 (carry 1): three equal values close as a ring and hand one unit to the next scale. In chemistry, this is a filled set of places — a complete shell.
• 1 + 1 = −1 (carry 1): two equal values leave tension behind. Tension is what a chemist calls reactivity.

On top of the carry sit three further principles. Closure: a loop only persists if it fits in a closed window of three passages; of the 27 possible windows, exactly 9 close. Memory: at every point in the network, the first passage fixes a direction and later passages are defined relative to it — the network carries its history as its shape. Scale: whatever fails to close at one scale becomes the raw material of the next, obeying the same rules again.

Finally, three directions with three values each form a cube of 27 cells. This cube turns out to be a surprisingly generous warehouse. Its corners, face centres and edge midpoints provide, ready-made, the exact bond angles of linear molecules (180°), flat molecules (120°), tetrahedral carbon (109.47°), octahedral molecules (90°) and close-packed crystals (60°). The eight corners split into exactly two interlocked tetrahedra that are mirror images of each other — and that single geometric fact will later explain why nature comes in left- and right-handed forms.

Nothing here looks like physics. No forces, no energies, no equations of motion. The claim is that the bookkeeping of the universe — what can close and what must remain open — already determines the architecture of matter.

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Part II — The atom without quantum mechanics

The most striking early result concerns the atom. Since Bohr (1913) and the quantum mechanics of the late 1920s, we explain the structure of atoms with Schrödinger’s equation and the peculiar geometry of electron orbitals. The book rebuilds the same structure from counting:

• A “set” of depth l has 2l + 1 places (the addresses −l to +l in balanced ternary). Each place holds two directions. Capacities: 2, 6, 10, 14 — which chemistry calls the s, p, d and f sets.
• Filling order follows from a simple cost: the number of inner rings plus the number of places. The resulting sequence is precisely the empirical order chemists have used since Madelung.
• A period ends where a particular set closes — a noble gas, standing in rest.

Run over all 118 elements, this counting puts every element in its correct period, predicts 96 of the 118 electron configurations exactly, and misses exactly the 22 elements where chemistry itself has long known the simple rule fails (chromium, copper, gold and their kin — the “early closure” exceptions). Where the configuration is right, the predicted number of unpaired electrons — which determines an atom’s magnetism — is right 84 times out of 84.

Even light is counted. A shell of depth n has n² places, so the binding per passage goes as 1/n²; a centre carrying Z passages binds Z times as strongly at Z times smaller depth, giving Z². Put together, this is Rydberg’s 1888 formula for spectral lines — the formula that first revealed the quantum structure of the atom. The ratios of hydrogen’s spectral wavelengths, which contain no units and can be tested directly, all agree with measurement to within a few hundredths of a percent. The same reading reproduces the ionisation energies of all ten one-electron ions, from hydrogen to neon, within 0.2%.

The honest caveat, which the book states clearly: the step from “places” to “binding strength” is a reading, an assumption, not a derivation. Once made, everything follows. This is the pattern of the whole book — a small number of interpretive bridges, and then a great deal of exact structure flowing across them.

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Part III — The bond, and the molecules Lewis missed

Gilbert Lewis’s 1916 picture of the shared electron pair still governs how chemistry is taught. It works beautifully — until it doesn’t. Three famous failures: oxygen (O₂) is magnetic, but Lewis’s rule says it should not be; carbon monoxide and nitric oxide do not fit his counting at all.

The book repairs all three by counting passages instead of bonds. Two atoms share places; each shared place can close (a meeting, low energy) or stay open (tension, high energy). A rule borrowed from the memory principle — later passages go the same way as the first — then reproduces the entire molecular-orbital theory of Hund and Mulliken: the magnetism of all twenty diatomic molecules tested, including the awkward ones, and the correct bond orders. Bond order, in turn, predicts bond length: of 39 pairs of molecules, 38 stand in the right order, and the ratio governing how much bonds shorten (log₂3 ≈ 1.585 against a measured 1.611) is Pauling’s 1947 law, here derived from the idea that deeper closures need logarithmically more depth.

A simple enumeration is even more striking. Define an atom’s “open passages” (hydrogen 1, carbon 4, nitrogen 3, oxygen 2, fluorine 1) and let a molecule be any connected set in which every passage closes. Generate everything with up to two heavy atoms: 24 formulas — water, methane, ammonia, hydrogen cyanide, hydrogen peroxide and friends — and all 24 exist. This is Lewis’s octet rule rediscovered from closure. (A caveat from the fact-check of this book: with fluorine included, the same rule also generates further real molecules such as CF₄ and NF₃, which the published list omits — an undercount rather than an error, but worth fixing.)

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Part IV — Shape and handedness

Why is carbon’s chemistry tetrahedral? Why do molecules come in mirror-image forms? Chemistry answers with VSEPR theory and stereoisomerism — empirical frameworks. The cube answers structurally: four directions around a carbon are the four alternate corners of the cube, at exactly 109.47°; the two possible tetrahedra are the cube’s two mirror forms, and no third form exists. A carbon bearing four different neighbours therefore carries a sign, + or −, and its mirror image carries the opposite. That is chirality — handedness — reduced to the geometry of three values.

The same cube offers exactly four neighbourhoods for solids — coordination numbers 4, 6, 8 and 12 — and 73 of 91 elemental crystal lattices choose one of them. Which one an element chooses remains open; the cube supplies the menu, not the order.

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Part V — Counting the world of carbon

Here the book stages its hardest test. Organic chemistry is the chemistry of carbon’s four open passages, and the deepest classic problem in the field is counting: how many distinct molecules have a given formula? The answers were computed by chemists and combinatorialists over 150 years, from Cayley’s trees (1875) to Pólya’s theory (1937), and are tabulated in the OEIS.

The book’s program generates every closed carbon tree and counts them, then counts mirror forms using the cube’s two tetrahedra — including the subtle case of the pseudo-asymmetric centre, where two branches differ only as mirror images of each other. The results are exact for every count:

• All alkanes up to 14 carbon atoms: 1, 1, 2, 3, 5, 9, 18, 35, 75, 159, 355, 802, 1858 — and with mirror forms 1, 1, 2, 3, 5, 11, 24, 55, 136, 345, 900, 2412, 6563. Every number matches the reference sequences exactly.
• All alcohols up to 10 carbons: 507 structures, 1553 with mirror forms — again exact at every step.
• Rings and double bonds from any formula (the “degree of unsaturation”): 14 of 14 correct, from benzene to caffeine to cholesterol to the 61 rings-and-bonds of buckminsterfullerene.
• Substituted benzenes: all 9 isomer counts exact — the ortho/meta/para pattern falls out of the hexagon’s twelve symmetries.
• Aromaticity: ring sets close at 2, 6, 10, 14, 18 passages — which is Hückel’s famous 4n+2 rule, with the identity 4n+2 = 2(2n+1) made visible. The aromatic ring obeys the same count as the atom.
• Acid strength: in oxyacids, the more oxygens share the leftover charge, the stronger the acid — Pauling’s second rule, with a rank correlation of −0.95 against 19 measured acids.

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Part VI — Life, from the same counting

The biochemical chapters are the most evocative. The same counting rules, applied to molecules of life:

• The sugars: glyceraldehyde 2 forms, the aldoses 4, 8, 16; the sugar alcohols 1, 3, 4, 10; tartaric acid exactly 3 (the two mirror forms and the meso form that Pasteur separated by hand in 1848). All exact, including the pseudo-asymmetric centres that simpler rules miss.
• The amino acids: all twenty proteins’ building blocks get the right number of mirror forms — glycine one (it is symmetric), isoleucine and threonine four (two centres each), the rest two.
• DNA: model each base’s edge as a word of three values — hydrogen-bond donor (+1), acceptor (−1), nothing (0) — and require pairs to meet without clashing, with a two-ring base always facing a one-ring base. Of all ten possible pairings, only A·T and G·C survive — precisely Watson and Crick’s 1953 result. Guanine turns out to be the exact negation of cytosine; the second DNA strand is the negation of the first.
• The genetic code: read as windows of three. Changing the second letter of a codon never preserves the amino acid (0.0% of cases); changing the third almost always does (71.6%). The second place fixes the address; the third is the remainder after the carry. (Independently confirmed during the fact-check of this book, digit for digit.)
• One hand only: life uses only L-amino acids and D-sugars. The memory rule explains propagation of a chosen hand — once a system has one, every later step copies it, as Soai’s famous self-amplifying reaction demonstrates in a flask. Why the first hand was chosen remains open.

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Part VII — Ice, and the pleasure of an exact check

Water ice keeps a disorder down to absolute zero: each hydrogen sits near one oxygen or the other, and the lattice does not decide for it. In 1935 Pauling estimated this residual entropy with a counting argument: of the 16 ways to orient four bonds around an oxygen, 6 satisfy the “ice rule” (two hydrogens near, two far). The entropy is R·ln(3/2) = 3.371 J per mole per kelvin, against the measured 3.41 ± 0.2 of Giauque and Stout.

The book’s move is to say: this is the closure rule — the same rule that filled the atom’s shells. And then it does something rare and satisfying: it tests the rule against an exact mathematical result. For the flat “square ice” lattice, the physicist Elliott Lieb computed the exact answer in 1967: (4/3)^1.5 = 1.53960 per site. The book’s transfer-matrix computation of the closure rule alone, on strips of growing width, converges to 1.53967 — a difference of 0.00007. An independent re-computation during the fact-check of this book reproduced the strip values to all five decimals.

This is the strongest single result in the book: a rule with no fitted parameters, reproducing a celebrated exact result of statistical mechanics.

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Part VIII — The balance: what is won, what is owed

Won. From one ground — three values, a carry, closure, memory, a cube — the book derives or reproduces: the capacities and filling order of atomic shells, the complete periodic system, atomic magnetism, the trend of ionisation energies and atomic sizes, the metal/non-metal dividing line, the octet rule, molecular-orbital theory with its magnetic oxygen, the bond-length law, molecular shapes, handedness, the exact counts of all alkanes and alcohols with their mirror forms, aromaticity, acid strength, the forms of the sugars and amino acids, the two base pairs of DNA, the structure and entropy of ice, and Lieb’s exact square-ice constant.

Owed. The book is candid about the ledger’s other side. Absolute numbers — energies in electronvolts, lengths in ångströms — are not derived; the theory works in ratios and needs one more law to fix its units. The 22 early-closure exceptions have no rule yet. The f-block of the periodic table (the lanthanides and actinides) resists, and one of its repairs is admittedly fitted. The lone pairs that bend water’s angle to 104.5° are not yet derived. Reaction rates are untouched.

The quantum computer question. The book began as a response to the claim that simulating molecules is the natural task of quantum computers. Its bounded answer: the qualitative core of chemistry — structure, counting, symmetry, stability — needs no wave function and no exponentially large state space. Every calculation here ran on an ordinary computer in seconds to minutes. Whether the quantitative core — precise energies — can also be captured by closure and carry is the open frontier, and the book does not pretend otherwise.

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Conclusion: how to read a book like this

Two attitudes are possible toward this work, and both are partly right.

The sceptic will note that “readings” — the interpretive steps that map chemical objects onto the net — are doing real work, that some are chosen with hindsight, and that reproducing known laws is easier than predicting new ones. All true, and the book itself says so.

But the fairer response is curiosity about what has actually been achieved. A single counting rule, with no adjustable parameters in most of its applications, lands the periodic table, molecular orbital theory, the exact combinatorics of organic chemistry, the code of life, and an exact result of statistical mechanics — each within a few percent, most within a fraction of a percent, several exactly. whatever one thinks of the metaphysics of the “vacuum net”, the arithmetic is real, checkable, and largely correct. A theory that can be checked by anyone with a laptop and that prints its own misses is doing what science, at its best, asks of a theory: making itself vulnerable.

The deepest suggestion of the book is not that quantum mechanics is wrong, but that it may be, in a precise sense, more than needed for chemistry’s architecture — that the qualitative skeleton of matter is a counting problem, visible before any equation of motion is written. Whether that suggestion ripens into new predictions — a rule for the early closures, a derivation of absolute energies — is the test the author has honestly set for himself.

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Annotated References

The references below combine the book’s own annotated list (verified bibliographically during the fact-check) with the primary sources against which its claims were checked. Annotations explain why each source matters.

The book under discussion

1. Konstapel, J. (2026). The Chemistry of the Vacuum.Net Theory. Constable Research, Leiden. — The work explained in this essay: 66 pages, eight parts, three appendices, and printed programs for every number. Its companion, The Vacuum.Net Theory, Ninth Edition (constable.blog, 2026), supplies the philosophical ground: trinity, carry, closure, memory, scale. The nilpotent algebra from which the vacuum.net descends is traced to Rowlands, P. (2007), Zero to Infinity: The Foundations of Physics (World Scientific) — a foundations-of-physics programme in which physical quantities combine into a nilpotent object; the “three values summing to zero” is its arithmetical echo.

The classical rules the theory re-derives

2. Lewis, G. N. (1916). “The atom and the molecule.” Journal of the American Chemical Society 38, 762–785. — The shared electron pair and the cubic atom. The book’s chapter on the bond is, by its own account, this paper re-grounded: Lewis drew atoms as cubes, and the vacuum.net gives the cube a derivation.
3. Madelung, E. (1936). Mathematische Hilfsmittel des Physikers. Springer. — Source of the empirical filling order (n + ℓ rule) that the book derives as a cost count of rings and places.
4. Hund, F. (1927). “Zur Deutung der Molekelspektren I.” Zeitschrift für Physik 40, 742–764. — The rule that equal places fill with parallel spins first; in the book this is the memory principle (N3) wearing chemical clothes.
5. Mulliken, R. S. (1932). “Electronic structures of polyatomic molecules and valence.” Physical Review 41, 49–71. — Molecular orbitals and bond order, the framework the book reproduces from the cube and the meeting rule. (Note: the book’s reference list cites volume 40; volume 41 is correct.)
6. Pauling, L. (1935). “The structure and entropy of ice…” JACS 57, 2680–2684; Pauling, L. (1947). “Atomic radii and interatomic distances in metals.” JACS 69, 542–553; Pauling, L. (1960). The Nature of the Chemical Bond, 3rd ed. Cornell. — Three Pauling laws re-derived here: the ice entropy R·ln(3/2), the logarithmic bond-order law (Q = log₂3), and the oxyacid strength rule.
7. Hückel, E. (1931). “Quantentheoretische Beiträge zum Benzolproblem.” Zeitschrift für Physik 70, 204–286. — The 4n+2 aromatic rule, shown in the book to be the atomic capacity count (2(2n+1)) applied to rings.
8. Gillespie, R. J. & Nyholm, R. S. (1957). “Inorganic stereochemistry.” Quarterly Reviews of the Chemical Society 11, 339–349. — VSEPR: the standard theory of molecular shape, to be compared with the cube’s directions.
9. Sanderson, R. T. (1967). Inorganic Chemistry. Reinhold. — The metal/non-metal boundary rule that the book’s “passages versus depth” reproduces (78 of 80 elements).

Spectra and the old quantum theory

10. Rydberg, J. R. (1890). “On the structure of the line-spectra of the chemical elements.” Philosophical Magazine 29, 331–337; Bohr, N. (1913). “On the constitution of atoms and molecules.” Phil. Mag. 26, 1–25. — The spectral formula the book re-derives from counting places (1/n² and Z²), and the first model that found the same exponents. (The book’s text dates Rydberg’s formula to 1888, his Swedish dissertation; the Philosophical Magazine summary is 1890.)
11. Kramida, A. et al. NIST Atomic Spectra Database. NIST. — The measured ionisation energies and ground-state terms against which the book’s atom chapters are scored.

Counting molecules: combinatorics meets chemistry

12. Cayley, A. (1875). “Über die analytischen Figuren, welche in der Mathematik Bäume genannt werden.” Berichte der deutschen chemischen Gesellschaft 8, 1056–1059. — The first count of alkanes as trees.
13. Pólya, G. (1937). “Kombinatorische Anzahlbestimmungen…” Acta Mathematica 68, 145–254. — The classic theory of counting under symmetry; both an ancestor and an independent check of the book’s tree-counting programs.
14. OEIS Foundation: sequences A000602, A000628, A000598, A000625 (oeis.org). — The reference counts of alkanes and alcohols with and without stereoisomers, all matched exactly (1858 and 6563 for C1–C14 alkanes; 507 and 1553 for C1–C10 alcohols).

Life and its structures

15. Pasteur, L. (1848). “Sur les relations…” Annales de Chimie et de Physique 24, 442–459. — The hand-separation of tartaric acid’s three forms, counted exactly by the book’s rules 176 years later.
16. Watson, J. D. & Crick, F. H. C. (1953). “Molecular structure of nucleic acids.” Nature 171, 737–738; Crick, F. H. C. (1966). “Codon–anticodon pairing: the wobble hypothesis.” J. Molecular Biology 19, 548–555. — The two base pairs and the wobble of the third codon position; the book derives the first from word-pairing and explains the second as a remainder after the carry.
17. Soai, K. et al. (1995). “Asymmetric autocatalysis…” Nature 378, 767–768. — Experimental proof that one hand can amplify itself in a flask; the book’s reading of the memory rule in biology.
18. Pauling, L., Corey, R. B. & Branson, H. R. (1951). “The structure of proteins…” PNAS 37, 205–211. — The α-helix and its i→i+4 hydrogen bonds, the motif the book reads as a closure of the protein chain with itself.

Ice and exact statistical mechanics

19. Giauque, W. F. & Stout, J. W. (1936). “The entropy of water and the third law of thermodynamics.” JACS 58, 1144–1150. — The measured residual entropy of ice (3.41 ± 0.2 J/mol·K) that the closure count hits at 3.371.
20. Nagle, J. F. (1966). “Lattice statistics of hydrogen bonded crystals. I.” Journal of Mathematical Physics 7, 1484–1491. — The exact series for real (hexagonal) ice: 3.409 J/mol·K — the target a future, non-independent count must reach.
21. Lieb, E. H. (1967). “Residual entropy of square ice.” Physical Review 162, 162–172. — The exact value (4/3)^1.5 = 1.53960 per site, reproduced by the book’s closure rule to five decimals — its strongest single result.

Element data and geometry

22. Pyykkö, P. & Atsumi, M. (2009). “Molecular single-bond covalent radii for elements 1–118.” Chemistry — A European Journal 15, 186–197. — The radii used in the size test. (One claim of the book — that fluorine’s radius slightly exceeds oxygen’s — does not match these values and merits correction.)
23. Kaupp, M. (2007). “The role of radial nodes of atomic orbitals for chemical bonding and the periodic table.” Journal of Computational Chemistry 28, 320–325. — The “primogenic effect”: why period-2 atoms differ from all others. The book derives the same flip from the carry of the inner ring.
24. Greenwood, N. N. & Earnshaw, A. (1997). Chemistry of the Elements, 2nd ed. Butterworth-Heinemann. — The standard reference for known oxidation states, the blind test of the book’s oxidation rules.
25. Mentel, L. M. mendeleev (github.com/lmmentel/mendeleev). — The open Python data package through which nearly all element data in the book’s programs were read — part of the book’s promise that every number can be re-derived by anyone.

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This essay is based on a full reading of the book and an independent recomputation of its checkable claims (arithmetic, spectral ratios, entropy values, transfer-matrix strips, isomer counts, base-pair logic, genetic-code statistics), of which essentially all matched exactly; the handful of discrepancies found — an incomplete molecule list, one radius claim, two reference details, and the treatment of noble gases in one table — are noted in the text and do not affect the book’s core results.

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