Curves in the Universe

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J. Konstapel,24 October 2026


© 2026 J. Konstapel, Constable Research, Leiden. All rights reserved. Quotation with
attribution is permitted; reproduction of the whole requires written permission.

Most theories begin with many ideas. The Vacuum.Net begins with one: the loop. A single strand runs through a set of places. Whenever the strand returns to a place it has already visited, a loop occurs. That return is the only event in the theory. Every other concept — path, knot, number, tension, scale — is derived from it.

The strand is not in space. The strand is the universe. There is nothing outside it. This is the first axiom. Every distinction we can make is itself a loop: to distinguish something is to come back to it a second time.

Five axioms fix the structure:

  1. Strand. There is exactly one strand. Every distinction is a loop.
  2. Closure. A loop persists only if it lies in a closed window. What does not close is passed on as tension.
  3. Memory. At every knot the passages are ordered. The first passage fixes the direction.
  4. Context. Every constant is a function of the strand.
  5. Scale. Merged into blocks, the coarse strand obeys the axioms again. What does not close at one scale is the material of the next.

This essay follows the consequences. What is a path? Where do paths connect? How do we compute a connection point? And why does the whole net never close?

2. Direction Needs Three Values

Direction is the first consequence. A loop has a direction. A direction must differ from its own reverse. This sounds trivial. It is not.

Mathematics offers a natural test. Which is the smallest cyclic group in which some element differs from its reverse? In the group with one element, every element is its own reverse. In the group with two elements, the nonzero element equals its reverse. Only the group with three elements — the integers modulo 3 — passes the test. There, +1 and −1 are distinct.

So direction has exactly three values. The theory calls them trits: +1, 0 and −1. A passage through a knot is read relative to the first passage through that knot. It either leaves the knot the same way (+1), leaves the way the first passage entered (−1), or leaves some other way (0). The rule uses only the strand itself. It is mechanical. Nothing is interpreted; everything is computed.

3. Closure Is a Loop of Trits

Trits behave like steps on the three-point cycle. Walk a sequence of trits and count your position modulo 3. The walk returns to its start exactly when the sum of the trits is 0 modulo 3. A closed run of trits is therefore itself a loop — one scale up. The scale axiom at work.

The shortest nontrivial return takes three steps: +1, +1, +1. This fixes the window at three trits. A window closes when its three trits sum to 0 modulo 3.

There are 27 possible windows. Only 9 close:

  • the three constant windows (000, +1+1+1, −1−1−1);
  • the six orderings of (−1, 0, +1).

The other 18 windows all have two equal trits and one different trit. None of them closes. Open windows are not harmless. By the second axiom, whatever does not close is passed on as tension. Tension is the engine of the theory. It guarantees that something always happens next.

4. Three Elementary Closures

The first theorem is the grammar of the theory. Every closed trit word reduces completely — to nothing — by repeatedly removing one of three pieces:

  • a standing: a single 0 (the path pauses);
  • a meeting: x followed by −x (the path goes and comes back);
  • a ring: x, x, x (the path goes round once).

No other word reduces. This was proven, and then computed for all 265,720 closed words up to length 12. Proof and enumeration agree.

Lift the steps to the integers and the picture sharpens. A closed walk with nonzero steps moves a multiple of three places. That multiple is its winding number. The meeting is the shortest closure with winding number 0. The ring is the shortest with winding number ±1. So apart from standing still, there are exactly two ways to close: to meet, in two steps, or to go round, in three steps. Everything else is a combination of these two moves.

5. Every Number Is a Path

The theory counts loops, so it must count. It counts in balanced ternary. Every integer has exactly one representation as a sum of digits −1, 0, +1 times powers of three. The word “balanced” is literal: the digits balance around zero.

  • Words of k digits reach exactly the integers from −(3^k − 1)/2 to +(3^k − 1)/2, with no gaps and no doubles.
  • Every number is a trit word; every trit word is a number.

Digit k stands at scale 3^k — the scale axiom again, now applied to arithmetic. The number of digits of an integer n is ⌊log₃(2n − 1)⌋ + 1, computable in whole numbers without any rounding. Depth in scales is therefore a logarithmic position: p = log₃ q. This position will carry the entire bridge to measurement in Section 8.

A few examples, writing T for −1: 2 = 1T, 3 = 10, 8 = 10T, −8 = T01.

6. The Ladder, and Why the Net Never Closes

Now merge the strand. At each scale, collect the elementary closures into blocks of equal size: a meeting merges two finest passages, a ring merges three. After a merges of meetings and b merges of rings, one coarse passage contains exactly 2^a · 3^b finest passages. This is the ladder theorem. The condition of equal block size is essential; drop it and the count fails (a ring joined to a single passage of a meeting gives 4, not 6).

The next theorem is the heart of the theory. Can the meetings and the rings ever return together? That would require 2^a = 3^b with a and b positive. This has no solution. A power of two is even; a power of three is odd. They meet only at 1, where a = b = 0.

The net never closes completely. There is always a remainder, and by the closure axiom the remainder passes on as tension. There is always a next passage. The strand has no last passage. The net is infinite — not by assumption, but by arithmetic.

Yet the net comes close to closing. It comes closest where b·log₂3 is nearest a whole number. These best approximations are the convergents of the continued fraction of log₂3. The sequence of near-closures runs at depths:

Depth baRemainder
12498.045 cents
23203.910 cents
5890.225 cents
121923.460 cents
416519.845 cents
53843.615 cents
3064851.770 cents
66510540.0756 cents
15601247270.0315 cents

The remainder at depth 12 is the Pythagorean comma — the small excess of twelve musical fifths over seven octaves, known to music theory for two millennia. Each near-closure is finite and enormous. A cycle at depth 665 holds about 1.9 × 10³¹⁷ finest passages.

7. What a Path Is

With the machinery in place, the definition is simple. A path is the strand on an interval: a contiguous piece of the one strand. Its trit word is the sequence of its trits, computed by the rule of Section 2.

A path is closed in a window when that window’s trits sum to 0. It is closed everywhere when every window closes. Its open windows are its tension. Two paths can meet only where the strand meets itself — at a knot.

The counts are exact:

  • A strand of N passages contains N(N + 1)/2 paths.
  • Of the 3^n possible trit words of length n ≥ 3, exactly 3^(n−1) close their first window.
  • Exactly nine words close everywhere. A word closes in every window if and only if its first window closes and it has period 3. There are nine closed windows, hence nine such words for every length: the constant runs and the repetitions of (−1, 0, +1) in its six orders.
  • A knot visited m times holds m(m − 1)/2 loops.

The nine everywhere-closed words carry a quiet message. A path without tension is possible — but only in nine rigid forms. Every other path carries tension. Tension is not the exception. It is the rule.

8. The Same Path in Position Space

The theory also lives in measurement. Take a phenomenon described by n magnitudes. Each magnitude is a ratio to a reference of its own kind, hence a pure number q. Its position is p = log₃ q — the depth of Section 5, now applied to data. The phenomenon is a point in an n-dimensional position space. When it changes, the point traces a path.

Between two magnitudes qᵢ and qⱼ along such a path, the local exponent is the ratio of their position changes. The slope trit is the direction of pⱼ relative to pᵢ: rising +1, flat 0, falling −1.

A point alone is a coordinate. A point carries no content. The path carries direction and shape. The point disappears; the path remains. This is the theory’s bridge from arithmetic to physics.

9. Connection Points on One Path

Where does a path connect with itself? Four answers, in the net and in position space.

In the net:

  • A knot — a place visited at least twice — holds m(m − 1)/2 loops.
  • A closed window — three trits summing to 0 — is one of the nine forms of Section 3.
  • An open window — any other three trits — carries tension.

A propagation theorem links neighbouring windows: the window sum shifts by the entering trit minus the leaving trit, sₖ₊₁ ≡ sₖ + tₖ₊₃ − tₖ (mod 3). Tension therefore does not stay put. It moves along the word.

In position space:

  • A regular point: the slope keeps its direction. Nothing closes.
  • A turning point: the slope changes direction. The trits read +1, 0, −1 (or reversed) — a meeting. The path goes and comes back.
  • A zero: the magnitude changes sign. The position p = log₃|q| runs to −∞. Again a meeting.
  • A return: the path passes a point it passed before — a knot at this level.

One theorem unifies the closed cases. On a closed path of a single magnitude, rises and falls alternate and are equal in number. The reduced slope word breaks down into meetings only, and the winding number is 0. Proven, and computed on 2,000 random periodic paths, then rechecked on 3,000. A closed path in one dimension is a chain of meetings. It cannot go round; there is nothing to go round in.

10. Connection Points Between Paths

Two paths are two pieces of the one strand. They meet where the strand meets itself: at a shared knot, where Section 9 applies unchanged.

A subtler connection is the seam. Join the end of one closed window A to the start of another closed window B. Two new windows straddle the joint. The seam rule counts the open ones: the number of open windows at the seam is [A₁ ≠ B₁] + [A₃ ≠ B₃]. The seam is closed if and only if B = A. A window continued in its own order produces no tension. Two different closed windows always produce tension at their seam.

Computed over all 81 ordered pairs: 9 closed seams, 36 with one open window, 36 with two. The pattern is exact and complete.

In position space, the between-path connections become laws. A power law qⱼ = C · qᵢ^e is a straight line in the plane of the positions, with slope e. Two such laws between the same magnitudes are identical, parallel, or meet in exactly one point:

RelationExponentsConstantsConnection
identicalequalequalthe same law
parallelequaldifferentsame exponent, different constant
intersectingdifferent—one point where both hold

And the no-closure theorem returns between two paths. Two periodic paths with frequencies f₁ and f₂ return together if and only if f₁/f₂ is rational. Twelve musical fifths never land on a whole number of octaves: 3¹² = 531,441 against 2¹⁹ = 524,288. Their ratio is the Pythagorean comma, 23.460 cents. The arithmetic of Section 6 is not an abstraction. It is audible in every piano tuned in fifths and played in octaves.

11. Computing a Connection Point

The method has four fixed steps. No step involves a choice.

  1. Strand. Take the input: a word sequence, a measured series, or a set of magnitudes.
  2. Trits. For a strand, apply the rule of Section 2. For magnitudes, use the slope trit. For an exact number, use its balanced-ternary word.
  3. Windows. Mark every window closed or open.
  4. Connection points. Read knots, seams, turning points, zeros and returns. Break every closure into standings, meetings and rings.

Five worked cases show the range:

CaseInputResult
A word sequencean 11-passage sentence4 knots, 6 loops, one window closed, one open
Lightwavelength against frequencyslope −1; λ·f constant; a straight line
Keplereight planets, NASA datameasured slope 1.4978 ± 0.0007 against the law’s 3/2
π junctiona superconducting junction on coolinga zero: a meeting
Little–Parksresistance of a ring against fluxa closed path: meetings only

The Kepler case deserves emphasis. The exponent 3/2 of the third law is exactly the ratio of a ring (length 3) to a meeting (length 2). The law of planetary motion reads as a balance of the two elementary closures. Whether this is coincidence or consequence is the kind of question the theory is built to ask.

12. Verification

Every theorem in the source has a proof. Every computed result was computed twice, by two independent programs: a respondent script, and a separate judge that wrote its own code from scratch without reading the first. The judge verified, among others:

  • the uniqueness of balanced ternary, by round trip on ±100,000;
  • the reduction of all 265,720 closed words up to length 12;
  • the seam rule over all 81 ordered pairs;
  • the nine everywhere-closed words for lengths 3 to 9;
  • the Kepler slope on NASA data, to five decimals;
  • the continued fraction of log₂3 and the depth table.

All 72 references were checked one by one against Crossref, publisher records or the source itself.

This discipline matters. A theory that claims the universe is a loop invites suspicion. Its response is to make every claim fixed, mechanical and checkable. Claims are fixed before the data are read; every number has two independent origins.

13. Conclusion: The Open Net

The Vacuum.Net yields a compact picture. Direction needs three values. Closure is arithmetic mod 3, and it comes in exactly three elementary forms: the standing, the meeting and the ring. Numbers are paths in balanced ternary, and their depth is logarithmic. Nine paths close everywhere; every seam between different closures carries tension.

And the whole never closes. Two to the a can never equal three to the b. The remainder is passed on as tension, forever. The net is infinite by a parity argument — the same argument that puts 23.46 cents between twelve fifths and seven octaves.

This is the theory’s most consequential claim. The universe is not a closed structure that happens to contain process. It is a process that keeps failing, by exactly a comma, to close — and each near-closure leaves a residue that becomes the material of the next scale. The curve in the Vacuum.Net does not end. It returns, misses by a remainder, and passes the difference on.

Annotated References

Grouped as in the source. Each entry carries a note on why it is cited.

The theory

  1. Konstapel, J. (2026). The Vacuum.Net Theory: Definition, Theorems and Tests, version 4. Constable Research, Leiden. — The founding document: the definition of the strand and the five axioms on which Sections 1 and 2 rest.
  2. Konstapel, J. (2026). “The Way Before the Net.” constable.blog, 20 October 2026. — Zero as the meeting of an act with its mirror; the two roads to closure (meeting and going round) behind Section 4.
  3. Konstapel, J. (2026). “The Mathematics of Closure.” constable.blog, 15 October 2026. — The convergents of log₂3 that generate the depth table of Section 6.
  4. Konstapel, J. (2026). “The Possibilities and Impossibilities of the Net.” constable.blog, 19 October 2026. — What the net can do in its own quantities: read a place, hold distant structures in one state, select by address, keep a configuration in circulation.
  5. Konstapel, J. (2026). The Respondent–Judge Protocol. Constable Research, Leiden. — The double-computation discipline verified in Section 12.

Numbers and the register

  1. Hardy, G. H., and Wright, E. M. An Introduction to the Theory of Numbers, 6th ed. Oxford University Press, 2008. — Unique factorisation, irrationality and continued fractions; the number theory behind the no-closure theorem.
  2. Knuth, D. E. The Art of Computer Programming, Vol. 2: Seminumerical Algorithms, 3rd ed. Addison-Wesley, 1998. — Balanced ternary as a practical register; Section 4.1.
  3. Spencer-Brown, G. Laws of Form. Allen & Unwin, 1969. — A calculus built on one act of distinction; the philosophical forerunner of the first axiom.
  4. Kauffman, L. H. (2005). “EigenForm.” Kybernetes 34(1/2), 129–150. — Forms that return on themselves: closure as a fixed point.
  5. Varela, F. J. (1975). “A calculus for self-reference.” International Journal of General Systems 2, 5–24. — A third value beside true and false, close to the trit 0.

Paths, returns and knots

  1. Pólya, G. (1921). “Über eine Aufgabe der Wahrscheinlichkeitsrechnung betreffend die Irrfahrt im Straßennetz.” Mathematische Annalen 84, 149–160. — A random path returns with certainty in one and two dimensions, not in three; the basis of the free-path model of the strand.
  2. Watson, G. N. (1939). “Three triple integrals.” Quarterly Journal of Mathematics 10, 266–276. — The integral behind the 3D return probability 0.3405.
  3. Montroll, E. W. (1956). “Random walks in multidimensional spaces.” Journal of the SIAM 4(4), 241–260. — Return probabilities in four and five dimensions.
  4. Rolfsen, D. Knots and Links. Publish or Perish, 1976. — The standard theory of closed curves in space and their winding.
  5. Adams, C. C. The Knot Book. W. H. Freeman, 1994. — A readable entry to knots, crossings and invariants.
  6. Milnor, J. Topology from the Differentiable Viewpoint. University Press of Virginia, 1965. — Degree and winding number; background to the elementary-closures theorem.

Curves, cycles and special points

  1. Strogatz, S. H. Nonlinear Dynamics and Chaos, 2nd ed. Westview Press, 2015. — Fixed points, limit cycles and quasi-periodic motion; background to the position-space sections.
  2. Lissajous, J. A. (1857). “Mémoire sur l’étude optique des mouvements vibratoires.” Annales de chimie et de physique (3rd series) 51, 147–231. — Combined vibrations that close only at rational ratios; the visual form of the two-cycles theorem.
  3. Thom, R. (1972). Stabilité structurelle et morphogenèse. Benjamin. — Folds and cusps: the stable kinds of sudden change in a curve.
  4. Arnold, V. I. (1992). Catastrophe Theory, 3rd ed. Springer. — A short, exact account of the special points of smooth maps.
  5. Feigenbaum, M. J. (1978). “Quantitative universality for a class of nonlinear transformations.” Journal of Statistical Physics 19, 25–52. — Period doubling: branching by factor two, one half of the ladder.
  6. Mandelbrot, B. B. (1982). The Fractal Geometry of Nature. W. H. Freeman. — Curves alike at every scale; background to the scale axiom.
  7. Verhulst, P.-F. (1838). “Notice sur la loi que la population suit dans son accroissement.” Correspondance mathématique et physique 10, 113–121. — The S-curve of bounded growth.

Scaling, laws and dimensions

  1. Buckingham, E. (1914). “On physically similar systems.” Physical Review 4, 345–376. — The Π theorem: laws written in dimensionless invariants; the basis of the pure-number positions.
  2. Bridgman, P. W. (1922). Dimensional Analysis. Yale University Press. — Why dimension is multiplicative.
  3. Barenblatt, G. I. (1996). Scaling, Self-Similarity, and Intermediate Asymptotics. Cambridge University Press. — Power laws as straight lines in logarithmic space, and when exponents are exact.
  4. Clauset, A., Shalizi, C. R., and Newman, M. E. J. (2009). “Power-law distributions in empirical data.” SIAM Review 51(4), 661–703. — How to fit an exponent honestly; required before any exponent is called exact.
  5. Stanley, H. E. (1971). Introduction to Phase Transitions and Critical Phenomena. Oxford University Press. — Critical exponents as straight stretches next to a threshold.
  6. Larson, D. B. (1959). The Structure of the Physical Universe. North Pacific Publishers. — Motion as the one component of the universe, with space and time as reciprocal aspects.
  7. Hestenes, D. (1966). Space-Time Algebra. Gordon and Breach. — Rotation as two reflections; the rotor that returns only after two turns.
  8. Grassmann, H. (1844). Die lineale Ausdehnungslehre. Wigand, Leipzig. — The extended form as the trace of a moving element: the strand as a map.

Music and the comma

  1. Barbour, J. M. (1951). Tuning and Temperament: A Historical Survey. Michigan State College Press. — The Pythagorean comma and the historical cycles of 12, 41 and 53 fifths.
  2. Benson, D. J. (2006). Music: A Mathematical Offering. Cambridge University Press. — The arithmetic of intervals, cents and temperaments.
  3. Kepler, J. (1619). Harmonices Mundi. Linz. — The third law, published in a book on musical ratios in the heavens.

Measurements used

  1. Williams, D. R. NASA, NSSDCA (updated 2025). Planetary Fact Sheet. — The distances and periods behind the Kepler case.
  2. Ryazanov, V. V., et al. (2001). “Coupling of two superconductors through a ferromagnet: evidence for a π junction.” Physical Review Letters 86(11), 2427–2430. — A critical current that passes through zero and changes sign: a zero in the sense of the theory.
  3. Little, W. A., and Parks, R. D. (1962). “Observation of quantum periodicity in the transition temperature of a superconducting cylinder.” Physical Review Letters 9, 9–12. — A closed curve in flux: the periodic worked case.
  4. Hertlein, C., et al. (2008). “Direct measurement of critical Casimir forces.” Nature 451, 172–175. — Critical Casimir forces near a critical point, with the 3D Ising exponent taken as given.

Method

  1. Tarski, A., Mostowski, A., and Robinson, R. M. (1953). Undecidable Theories. North-Holland. — What it means to translate one theory into another.
  2. Lakatos, I. (1970). “Falsification and the Methodology of Scientific Research Programmes.” In Criticism and the Growth of Knowledge, Cambridge University Press, 91–196. — Why claims must be fixed before the data are read.

Topology of closed curves

  1. Gauss, C. F. (1833). Note of 22 January 1833 on the linking of two closed curves. Werke, Vol. 5, p. 605. — The linking number: the first invariant of two curves.
  2. Alexander, J. W. (1928). “Topological invariants of knots and links.” Transactions of the AMS 30(2), 275–306. — The first polynomial that distinguishes knots.
  3. Jones, V. F. R. (1985). “A polynomial invariant for knots via von Neumann algebras.” Bulletin of the AMS 12(1), 103–111. — A sharper knot invariant, later linked to physics.
  4. White, J. H. (1969). “Self-linking and the Gauss integral in higher dimensions.” American Journal of Mathematics 91(3), 693–728. — Linking = twist + writhe; the split of a closed ribbon’s winding, used for DNA.

Cycles, rotation and locking

  1. Poincaré, H. (1885). “Sur les courbes définies par les équations différentielles (3e partie).” Journal de Mathématiques Pures et Appliquées (4th series) 1, 167–244. — The rotation number; a curve on a torus closes iff that number is rational. The exact form of the two-cycles theorem.
  2. Kronecker, L. (1884). “Näherungsweise ganzzahlige Auflösung linearer Gleichungen.” Sitzungsberichte der Akademie der Wissenschaften zu Berlin, 1179–1193 and 1271–1299. — At an irrational ratio the curve never closes but comes arbitrarily close to every point.
  3. Weyl, H. (1916). “Über die Gleichverteilung von Zahlen mod. Eins.” Mathematische Annalen 77(3), 313–352. — Multiples of an irrational spread evenly; background to the remainder table.
  4. Arnold, V. I. (1961). “Small denominators. I.” Izvestiya Akad. Nauk SSSR 25(1), 21–86. — Arnold tongues: coupled cycles lock onto a rational ratio over a finite range; closure becomes stable, not exceptional.
  5. Kuramoto, Y. (1975). “Self-entrainment of a population of coupled non-linear oscillators.” Lecture Notes in Physics 39, Springer, 420–422. — Many cycles fall into one common rhythm above a coupling threshold.
  6. Bennett, M., Schatz, M. F., Rockwood, H., and Wiesenfeld, K. (2002). “Huygens’s clocks.” Proceedings of the Royal Society A 458, 563–579. — Huygens’ 1665 observation repeated: two clocks on one beam lock in anti-phase; a measured two-cycle with coupling.
  7. Noether, E. (1918). “Invariante Variationsprobleme.” Nachrichten der Gesellschaft der Wissenschaften zu Göttingen, 235–257. — Every continuous symmetry gives a conserved quantity; the physical source of the constants in the two-laws theorem.

Physics: closed paths and whole numbers

  1. Thomson, W. (Lord Kelvin) (1869). “On vortex atoms.” Proceedings of the Royal Society of Edinburgh 6, 94–105. — Atoms as knotted closed curves in a medium; the historical forerunner of the knot in the net.
  2. Aharonov, Y., and Bohm, D. (1959). “Significance of electromagnetic potentials in the quantum theory.” Physical Review 115(3), 485–491. — A particle on a closed path picks up a phase from a field it never touches; only the closed path counts.
  3. Deaver, B. S., and Fairbank, W. M. (1961). “Experimental evidence for quantized flux in superconducting cylinders.” Physical Review Letters 7(2), 43–46. — Flux in a superconducting ring comes in whole units of h/2e.
  4. Doll, R., and Näbauer, M. (1961). “Experimental proof of magnetic flux quantization in a superconducting ring.” Physical Review Letters 7(2), 51–52. — The same result, measured independently in the same month.
  5. Josephson, B. D. (1962). “Possible new effects in superconductive tunnelling.” Physics Letters 1(7), 251–253. — Current across a junction depends on phase difference; the basis of every junction case.
  6. Bulaevskii, L. N., Kuzii, V. V., and Sobyanin, A. A. (1977). JETP Letters 25, 290. — The theoretical prediction of the π junction, confirmed 24 years later.
  7. Berry, M. V. (1984). “Quantal phase factors accompanying adiabatic changes.” Proceedings of the Royal Society A 392, 57–72. — A system led round a closed loop returns with a phase set by the loop’s geometry.
  8. von Klitzing, K., Dorda, G., and Pepper, M. (1980). “New method for high-accuracy determination of the fine-structure constant based on quantized Hall resistance.” Physical Review Letters 45(6), 494–497. — A resistance that takes exact values h/(ie²): a measured whole-number closure.
  9. Thouless, D. J., et al. (1982). “Quantized Hall conductance in a two-dimensional periodic potential.” Physical Review Letters 49(6), 405–408. — The integer of the Hall effect is a winding number; topology explains the exact value.
  10. Witten, E. (1989). “Quantum field theory and the Jones polynomial.” Communications in Mathematical Physics 121(3), 351–399. — Knot invariants from a quantum field theory; knots and physics in one formalism.
  11. Faddeev, L., and Niemi, A. J. (1997). “Stable knot-like structures in classical field theory.” Nature 387, 58–61. — A field theory whose stable particles are knots.
  12. Kleckner, D., and Irvine, W. T. M. (2013). “Creation and dynamics of knotted vortices.” Nature Physics 9, 253–258. — Knotted vortices made in water and filmed; Kelvin’s idea tested.
  13. Rowlands, P. (2007). Zero to Infinity: The Foundations of Physics. World Scientific. — Physics built from a total of zero; the nilpotent condition as a closure condition.

Chemistry: chirality, order and molecular knots

  1. Pasteur, L. (1848). “Mémoire sur la relation qui peut exister entre la forme cristalline et la composition chimique.” Comptes rendus 26, 535–538. — Left- and right-handed crystals of one compound; the chemical counterpart of a curve’s direction.
  2. Mendelejeff, D. (1869). Zeitschrift für Chemie 12, 405–406. — The periodic system: properties returning in periods of 2, 8, 8, 18, 18, 32, 32 — a closure pattern in chemistry.
  3. Frisch, H. L., and Wasserman, E. (1961). “Chemical topology.” Journal of the American Chemical Society 83(18), 3789–3795. — The programme for molecules that are knots and links.
  4. Dietrich-Buchecker, C. O., Sauvage, J.-P., and Kintzinger, J.-P. (1983). “Une nouvelle famille de molécules : les métallo-caténanes.” Tetrahedron Letters 24(46), 5095–5098. — The first efficient synthesis of two interlocked rings: linking number 1.
  5. Dietrich-Buchecker, C. O., and Sauvage, J.-P. (1989). “A synthetic molecular trefoil knot.” Angewandte Chemie Int. Ed. 28(2), 189–192. — The first molecule tied as a trefoil knot.
  6. Anelli, P. L., Spencer, N., and Stoddart, J. F. (1991). “A molecular shuttle.” Journal of the American Chemical Society 113(13), 5131–5133. — A ring moving between two stations on a thread; a mechanical bond used as a switch.
  7. Koumura, N., et al. (1999). “Light-driven monodirectional molecular rotor.” Nature 401, 152–155. — A molecule that turns one way only; chirality sets the direction of a closed cycle.
  8. Taylor, W. R. (2000). “A deeply knotted protein structure and how it might fold.” Nature 406, 916–919. — A natural protein whose chain forms a knot: knotted curves in living matter.

All 72 entries were checked one by one against Crossref, publisher records or the source itself by a separate judge. Source: J. Konstapel, “Curves in the Vacuum.Net”, v1.0, Constable Research, Leiden, 24 October 2026. Quotation with attribution is permitted.

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