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Veronese, Cremona, and the Mystical Hexagram

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Algebraic Geometry between Tradition and Future (INdAM 2021)

Part of the book series: Springer INdAM Series ((SINDAMS,volume 53))

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Abstract

Apart from a brief historical introduction on what precedes them, this chapter limits itself to presenting the two almost simultaneous works (1877) of the young Veronese and the already famous Cremona in the belief that they constitute significant moments for both the scholars. Pascal mysticum hexagram, after the works of Steiner, Kirkman, Plücker, Cayley, and Salmon, provided a complex configuration made by 60 Pascal lines (one for each of the hexagons built on the same points of a conic), which converge 3 by 3 on Kirkman’s 60 points (K) and Steiner’s 20 points (S). The points K lie 3 by 3 on the Pascal lines p. Each line p passes through a point S; each of the 15 Plücker (pl) lines passes through 4 points S. The points K lie 3 by 3 in the Cayley lines (c); the lines c pass through 3 of the 15 Salmon’s points (Sa); every line c passes through a point S; 4 lines c pass through each Sa point. In his paper, Veronese added three new results: It is possible to divide the complex of the mystical hexagram into 6 configurations, each with 10 Pascal lines and 10 Kirkman points, plus 10 Steiner points, shared with two other configurations; two different configurations always have in common 4 Steiner points and a Plücker line that joins them; it is possible to obtain infinite configurations that are isomorphic with the first (Conway and Ryba’s multi-mystic). Cremona showed that the same results could be obtained from the configuration of 15 lines and 15 tritangent planes on a cubic surface (the Cremona–Richmond configuration). My chapter deals with the two different approaches to the two mathematicians.

Translated by Denis Gailor

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Notes

  1. 1.

    Here and afterwards, the English translation of the Italian citations is of the author. Siccome io dovevo preparare un lavoretto per tenere una conferenza tra i miei amici studenti del Politecnico di Zurigo nel Mathematische Seminar diretto dagli illustri sigg. Prof. Fiedler e Frobenius, nel giugno dell’anno testé passato, così mi proposi allora di risolvere la questione di Hesse e Schröter, e poiché io credo di averla non solo completamente risoluta, ma ben anco avervi aggiunti altri teoremi importanti, così mi faccio animo di presentare ora questo mio piccolo lavoro ai sigg. Professori dell’Università Romana, con la speranza che sia ben accettato [6, p. 651].

  2. 2.

    Beltrami had also supported Veronese’s requests. In a letter dated 4 October 1876, he wrote: He is a talented young man, especially in pure geometry, and I think he is perfectly eligible for the fourth year. (Biblioteca dell’Istituto Mazziniano di Genova, Fondo Itala Cremona Cozzolino, Sc. 49, 9992).

  3. 3.

    Veronese’s and Cremona’s letters to Fiedler can be found in [7], where there are also summaries of the ones to Cremona, the full texts of which can be found in [8].

  4. 4.

    Many nineteenth-century commentators attribute to Steiner the idea that Pascal’s theorem refers to any of the 60 permutations. It does not seem to me in any way that such an idea was not already present in Pascal himself.

  5. 5.

    The rule for switching from the pairs of triangles to the hexagon is simple: just start from any side of the first and then continue alternating between the first and the second. Thus, AB is followed by BE, then EF, then FD, and then DC. Conversely, the triangles are obtained from the hexagon by alternating the sides. Thus, AB will be the first side of the first triangle and BE that of the second. There will, respectively, follow EF and FD and lastly DC and CA.

  6. 6.

    [13], Anhang, 54–58, pp. 311–313.

  7. 7.

    I use the term “amateur” only in the sense of “not belonging to the academic world.” On the biography of this interesting mathematician, I refer the reader to [14], also recalling that the author is Dikran Tahta, Stephen Hawking’s teacher.

  8. 8.

    As already noted, it was Hesse’s conjecture that prompted Veronese to begin his work.

  9. 9.

    The triangles are presented in an orderly way, in the sense that the first, second, and third sides correspond to each other.

  10. 10.

    Indeed, M = (BE, CF) belongs to p(ABEDCF) and the same goes for N and O.

  11. 11.

    The three triangles are obtained in the only way in which it is possible to use in one way the nine sides of the complete hexagon not used in the first two.

  12. 12.

    Christine Ladd (1847–1930) is best known for her contributions to psychology. In 1876, at the request of Sylvester himself, she had enrolled at Johns Hopkins University using, in presenting her credentials, the name C. Ladd, which did not specify her gender. On the difficulties encountered in relation to the University’s anti-female regulations, I refer to [16]. Here, it suffices to recall that in 1882, despite having regularly submitted the documents for the PhD, she was not allowed to obtain it. With a belated recognition of its error, the University conferred the qualification on her in 1926, when she had already turned 78! She was also a student of Sanders Peirce, who praised her studies on Boolean algebra.

  13. 13.

    So, with this symbol, I mean the line joining the two Kirkman points obtained using the two permutations that have A, B as the first and second elements and D, E as the fourth and fifth and the remaining ones C, F in the two possible positions. Thus, (BC; EF) refers to the line between K(BCAEFD) and K(BCDEFA). The lines (AB; DE) and (DE; AB) coincide, as is easily verified.

  14. 14.

    Perhaps, it is worth pointing out that here I have limited myself to a copy-paste of what was previously written about the first level. Perhaps, it is appropriate here to quote Poincaré saying that mathematics is “l’art de donner le même nom à des choses différents.”

  15. 15.

    I have slightly modified the arrangement of the vertices of the hexagon to make the figure more readable.

  16. 16.

    By chance, I wrote this on April 18, 2021, a few days after the first anniversary of the death of John Conway (April 11, 2020), the recollection of whom, as a mathematician who was in many ways extraordinary and completely unusual, remains indelible.

  17. 17.

    Avendomi il sig. Veronese pregato di leggere la sua Memoria, io feci pensiero di verificare i risultati in essa contenuti per una via diversa da quella che l’A. aveva seguita. Mentr’egli si è attenuto sempre alla geometria piana, io ebbi ricorso allo spazio a tre dimensioni e propriamente ad una superficie di terzo ordine dotata di un punto doppio, ed ottenni così delle figure che proiettate dal punto doppio somministrano immediatamente quella del sig. Veronese [Transunti della R. Accademia dei Lincei (3),1, seduta dell’8 aprile 1877, p. 146].

  18. 18.

    The reader is reminded that not many years earlier, in 1866, Cremona had won the Steiner prize with a profound memoir [24] precisely on cubic surfaces. On them, and also on the memoir cited, see [25].

  19. 19.

    Of course, this is equivalent to saying that the corresponding tritangent planes intersect in a line not lying on the surface. Dolgachev calls the pair of tritangent planes thus obtained a Cremona pair, attributing this nomenclature to Reye.

  20. 20.

    For example, it is enough to replace the word “triplet” of triangles with “trihedron” to pass identically from one construction to another. It will in fact be noted that the figures constructed for the projections correspond exactly to those created directly in the plane.

  21. 21.

    For example, the Plücker line of π1 intersects that of the plane π2 at the Steiner point (ACE; BDF), as can be easily verified from the table.

  22. 22.

    In the foregoing, I have called these lines Plücker lines.

  23. 23.

    Nelle cose esposte sinora, l’ipotesi di una superficie Φ di terz’ordine dotata di un punto conico O non è stata necessaria se non in quanto s’è voluto avere un centro di projezione per dedurre dalle proposizioni stereometriche i teoremi planimetrici dell’hexagrammum mysticum. Che se si prescinde da tale projezione, le proprietà dimostrate nella prima parte di questa Memoria presuppongono unicamente l’esistenza di un sistema di quindici rette situate tre a tre in quindici piani. I quindici piani si aggruppano in sei pentaedri i cui vertici e i cui spigoli ho chiamati punti di Kirkman e rette di Pascal; e si aggruppano inoltre in venti triedri conjugati due a due, i cui vertici sono i punti di Steiner: punti situati quattro a quattro in quindici rette, dette rette di Steiner. Le rette ed i punti di Steiner sono spigoli e vertici di un esaedro, che costituisce in certo modo il nucleo dell’intera figura ed alle cui sei facce sono coordinati i sei pentaedri [23 , p. 866].

  24. 24.

    l’insieme dei loro spigoli e de’ loro vertici è analogo al sistema delle sessanta rette di Pascal e de’ sessanta punti di Kirkman [23 , p. 871].

  25. 25.

    A similar succession is obtained in [26] (where it is called a Veronese sequence).

  26. 26.

    Per ottenere precisamente le figure … considerate dal giovane geometra … basta partire dai pentaedri k = 1, dedurre da esso il conjugato nell’involuzione k + k’ = 4, poi da questo il coniugato nell’involuzione k’k” = 1, indi da quest’ultimo il conjugato nell’involuzione k” + k”’ = 4, e così via di seguito, alternando le due involuzioni indefinitamente [23 , p. 874].

  27. 27.

    As is well known, today this configuration (15)3 is called the Cremona-Richmond one [4, 5]. The name of Richmond derives from the hyperspace reinterpretation made by the American mathematician of Cremona’s construction in [27].

  28. 28.

    Letter from Cremona to Hirst of August 24, the Italian original in [29 , p. 181], reads: Sono ora occupato intorno al problema di determinare il pentaedro d’una superfice del 3° ordine della quale siano date le 27 rette. In sostanza ho già ottenuta la soluzione geometrica, deducendola da certi teoremi recenti di Reye sui pentaedri ed esaedri polari … coi risultati da me ottenuti colla trattazione stereometrica dell’hexagrammum mysticum (A proposito della memoria del Sig. Veronese).

  29. 29.

    Letter from Cremona to Hirst of November 7, 1877, Io feci una piccola comunicazione alla Sezione Matematica circa la soluzione del problema di ridurre l’equaz.e di una superficie generale del 3° ordine alla forma

    $$ \sum_{r=1}^{r=6}{x}_r^3=0\ sotto\ la\ condizione\sum {x}_r=0 $$

    È un risultato al quale pervenni mentre mi trovavo nelle Alpi … in continuazione delle ricerche contenute nel piccolo lavoro stampato dai Lincei [29 , p. 183].

  30. 30.

    This is the famous and much-cited hexahedral equation of Cremona. For a modern treatment of the subject, see [25].

  31. 31.

    Ti ricorderai che quando mi fu offerto l’attuale ufficio in Roma io non accettai senza esitazione … Prevedevo di dovermi sobbarcare a occupazioni assai gravi, ma non mi sarei mai sognato un sacrificio così enorme e completo. Assorbito tutto il mio tempo, tutte le mie forze da lavori amministrativi, non potei più far nulla per la scienza, nella quale è riposta l’unica mia ambizione [Letter to his wife of July 29 1877, Lascito Itala Cremona Cozzolino–Archivio dell’Istituto mazziniano di Genova, 051–11,831].

  32. 32.

    In questo luogo fresco e salubre dove l’occhio ammira le gigantesche guglie dolomitiche e si riposa sul verde de’ prati e de’ boschi estesissimi, dove non giunge alcun’eco de’ fastidi romani, io conduco una vita veramente felice. Il tempo diviso tra le passeggiate e le escursioni, insieme con la mia famiglia, e il lavoro scientifico, scorre semplicemente, serenamente - in modo assai diverso da quello che mi accadeva in Roma. E il presente mi riesce tanto più dolce perché spero nell’avvenire (Letter to Hirst, August 24 1877, in [29 , p. 181]).

  33. 33.

    Arrivando a Roma … ebbi assalti formidabili da Blaserna, Battaglini, Cannizzaro e da parecchi professori della scuola (Letter of Cremona to Enrico Betti, October 8 1877, in [30 , p. 79]).

  34. 34.

    Ora o mai. Sono ormai ad una età in cui non c’è più tempo da perdere. Se tardassi ancora qualche anno il ritorno agli studj mi sarebbe impossibile e non resterebbe di me che un cattivo arnese burocratico [Letter to his wife, July 29, 1877 already quoted].

  35. 35.

    Introdurre nella geometria un nuovo gruppo di trasformazioni, non imposte artificialmente, ma imposte dalla natura stessa vuol dire, in primo luogo, offrire il mezzo di trasportare proprietà note di enti semplici ad enti più complessi … Ma vuol dire in secondo luogo dar origine allo studio di quelle proprietà geometriche che non vengono alterate dalle trasformazioni stesse. Di queste due parti del programma che la scoperta del Cremona permetteva di formulare egli svolse la prima; … La seconda parte del programma … ha ispirato le principali ricerche che nel campo della geometria algebrica furono condotte nell’ultimo cinquantennio … È difficile dire se egli ne prevedesse i successivi sviluppi, ma posso affermare, anche per graditi ricordi personali che egli ne seguì i progressi col più grande interesse, fino agli ultimi giorni della sua vita [31].

  36. 36.

    On the “coldness” with which Cremona received Bertini’s important results of 1877, I refer to [32].

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  1. Dipartimento di Matematica e Informatica, Palermo, Italy

    Aldo Brigaglia

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  1. Department of Mathematics and Computer Science, University of Palermo, Palermo, Italy

    Gilberto Bini

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Brigaglia, A. (2023). Veronese, Cremona, and the Mystical Hexagram. In: Bini, G. (eds) Algebraic Geometry between Tradition and Future. INdAM 2021. Springer INdAM Series, vol 53. Springer, Singapore. https://doi.org/10.1007/978-981-19-8281-1_15

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