Skip to content
DraconiticResearch
Search Sources API
193,349 studies31 sourcesUpdated 15 min ago
PreprintCC BY 4.0arXiv

Descartes' Circle Theorem, Princess Elizabeth, and Spinors

  • Daniel Parrochia
Open at sourcePDFDOICite

Abstract

In this article, on Descartes' famous "circle theorem," we first attempt to explain how the philosopher could have arrived at the equation that he presents without demonstration in a letter to Princess Palatine Elizabeth of Bohemia in November 1643 (a previously unpublished proof, to our knowledge). Then we report some stages of the subsequent generalization of this theorem, whose Cliffordian flavor, via the equivalence of a quadratic form and the square of a linear form, still mobilizes mathematicians today. This statement, which, over time, has undergone different extensions, Euclidean and non-Euclidean, has found, {\it in fine}, a spinor formalization. We see there the proof of what we may call, in a Bachelardian style, an "inductive value" of the truth, which extends by successive generalizations. We conclude, more briefly, with the contacts between Elizabeth and Descartes, and the perhaps symbolic meaning of these considerations on "kissing circles," as they are called in Anglo-Saxon countries, in the context of their correspondence on the soul and the body, and the question of passions.

Related

  1. Random domino tilings and the arctic circle theorem
    William Jockusch, James Gary Propp, Peter W. Shor
    Annales de l’Institut Henri Poincaré D Combinatorics Physics and their InteractionsPeer-reviewedCC BY224 citedMathematical physicsOpenAlex
  2. Generalized Killing spinors as Lagrangian sections of principal spin bundles
    Fumihiro Ueno
    arXivPreprintCC BY 4.0Full textGeometry & topologyarXiv
  3. Descartes' rule of signs for arbitrary fewnomial systems
    Frédéric Bihan
    arXivPreprintarXiv licenceFull textAlgebra & number theoryCombinatorics & discrete mathematicsarXiv
  4. Tournaments, flags, and spinors
    Diego Conti, Simon Salamon
    arXivPreprintCC BY 4.0Full textGeometry & topologyarXiv
  5. The universal moduli space of non-degenerate ℤ/2ℤ harmonic spinors
    Siqi He, Gregory J. Parker, Thomas Walpuski
    arXivPreprintCC BY-NC-SA 4.0Full textGeometry & topologyarXiv

Cite

BibTeXRIS
BibTeX
@misc{parrochia2026descartes,
  title = {{Descartes' Circle Theorem, Princess Elizabeth, and Spinors}},
  author = {Daniel Parrochia},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.22388},
  eprint = {2609.22388},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.22388},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - Descartes' Circle Theorem, Princess Elizabeth, and Spinors
AU  - Daniel Parrochia
PY  - 2026
DA  - 2026/09/30
PB  - arXiv
DO  - 10.48550/arxiv.2609.22388
UR  - https://arxiv.org/abs/2609.22388
AB  - In this article, on Descartes' famous "circle theorem," we first attempt to explain how the philosopher could have arrived at the equation that he presents without demonstration in a letter to Princess Palatine Elizabeth of Bohemia in November 1643 (a previously unpublished proof, to our knowledge). Then we report some stages of the subsequent generalization of this theorem, whose Cliffordian flavor, via the equivalence of a quadratic form and the square of a linear form, still mobilizes mathematicians today. This statement, which, over time, has undergone different extensions, Euclidean and non-Euclidean, has found, {\it in fine}, a spinor formalization. We see there the proof of what we may call, in a Bachelardian style, an "inductive value" of the truth, which extends by successive generalizations. We conclude, more briefly, with the contacts between Elizabeth and Descartes, and the perhaps symbolic meaning of these considerations on "kissing circles," as they are called in Anglo-Saxon countries, in the context of their correspondence on the soul and the body, and the question of passions.
ER  -