# Imperial Porphiry and Golden Leaf: Sierpinski Triangle in a Medieval Roman Cloister

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Part of the Advances in Intelligent Systems and Computing book series (AISC,volume 809)

## Abstract

In medieval churches motives are found, similar to what we call today “Sierpinski triangle”: a same composition of full and void areas, interweaved and repeated at smaller and smaller scale. The motive has seen its mathematically rigorous definition in 1915, and has been a “benchmark” for scientists thereafter. Mathematicians imagine and study what would remain upon carrying on indefinitely the procedure of inserting voids: a “powder of points” would be left, organized in a precise way around large voids. On the other hand, the geometrical compositions of the Marmorari Romani, loosely known as “Cosmati”, are often characterized by repetitions on different spatial scales, thus suggesting to the spectator an in-depth view (Moran and Williams, Boll. Mat. Ital. Sez. A, ser. VIII, VII-A:17–47 (2004), [22, Williams in Math. Intell. 19:41–45 (1997), 30]). Actual artifacts composed iteratively can be reliably analyzed with mathematical methods, if the motive shows at least three levels of iteration (i.e. different spatial scales). In Conversano and Tedeschini Lalli Aplimat (J Appl Math 4:113–122, [10]) we reviewed some such triangles, in stone, isolated in medieval floors. In the present paper we report about some new examples recently found in Rome, and not published yet in any form. These are isolated Sierpinski triangles in golden leaf, contained in the frieze of medieval cloisters. Their composition is iterated at 3 and 4 levels. Moreover their placement warrants to their authenticity. Where the stones have fallen out, empty lodgings along the frieze testify that it has gone untouched for a long time. The “protagonists” of the motive are the smallest stones, as they are the ones in golden leaf, i.e. all that is perceived when looking up. Moreover the golden leaf reflects the light, adding the perceptually smaller spatial scales of the shimmering.

### Keywords

• Marmorari Romani
• Sierpinski triangle
• Cosmatesque art
• Visual perception

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## References

1. Alligood, K.T., Tim, D., Sauer, T.D., Yorke, J.A.: Chaos: An Introduction to Dynamical Systems. Springer, New York (1996)

2. Bellini, M., Brunori, P.: Nota sull’ipotesi ricostruttiva dell’ambone di Giovanni VII in Santa Maria Antiqua. In: Santa Maria Antiqua tra Roma e Bisanzio. cat. of exhibition, Electa, Roma (2016), pp. 13–133

3. Benevolo, L.: Una statistica sul repertorio geometrico dei Cosmati. In: Quaderni dell’Istituto di Storia dell’Architettura, 5, pp. 11–20 (1954)

4. Brunori, P., Carboni, F.: Il disegno come strumento. Rilievi, ricostruzioni, modelli nello studio del fregio. In: De Strobel, A.M. (ed.) Il portico medievale di San Giovanni in Laterano. I frammenti ritrovati. Edizioni.Musei Vaticani, Città del Vaticano, pp. 165–187 (in print)

5. Cantor, G.: Über unendliche, lineare Punktmannigfaltigkeiten V. Math. Ann. 21, 545–591 (1883), see also Volterra, V.: Alcune osservazioni sulle funzioni punteggiate discontinue. G. Mat. 19, 76–86 (1881)

6. Chabert, J.L.: The early history of fractals. In: Grattan-Guinness, I. (ed.) Companion Encyclopedia of the History and Philosophy of the Mathematical Sciences, pp. 367–374. Roudtledge, London and NY (1992)

7. Clausse, G.: Les monuments du christianisme au moyen-âge, p. 429. Ernest Leroux Éditeur, Paris (1897)

8. Claussen, P.C.: Magistri Doctissimi Romani: die römischen Marmorkünstler des Mittelalaters. Corpus Cosmatorum. Franz Steiner Verlag, Stuttgart (1987)

9. Claussen, P.C.: Die Kirchen der Stadt Rom im Mittelalter 1050–1300. Vol. 2: San Giovanni in Laterano. Franz Steiner Verlag, Stuttgart (2008)

10. Conversano, E., Tedeschini Lalli, L.: Sierpinsky triangles in stone, on medieval floors in Rome. Aplimat J. Appl. Math. 4, 113–122 (2011)

11. Dauben, W.D.: Set theory and point set topology. In: Grattan-Guinness, I. (ed.) Companion Encyclopedia of the History and Philosophy of the Mathematical Sciences, pp. 351–359. Roudtledge, London and NY (1992)

12. Del Bufalo, D.: L’Università dei Marmorari di Roma. L’Erma di Bretschneider, Roma (2007)

13. Del Bufalo, D.: Marmorari Magistri Romani. L’Erma di Bretschneider, Roma (2010)

14. Enriques, F.: Questioni riguardanti le matematiche elementari. Zanichelli, Bologna (1912)

15. Falcolini C., Tedeschini Lalli, L.: Compounds of helical curves: medieval twisted columns. In: Proceedings of the 15th Conference on Applied Mathematics, Aplimat 2016, pp. 324–332. Slovak University of Technology, Bratislava, Slovacchia (2016)

16. Giovannoni, G.: Opere dei vassalletti marmorari romani. In: L’Arte, XI, f.4, pp. 262–283 (1908)

17. Jones, O.: The grammar of ornament. Day and Son, London (1856)

18. Krautheimer, R.: Rome: profile of a city, 312–1308. Princeton University Press, Princeton (1980)

19. Lombardo Radice, L.: L’infinito. Editori Riuniti, Roma (1981)

20. Maire Viguer, J.: L’altra Roma. Una storia dei romani all’epoca dei comuni (secoli XII-XIV). Torino, Einaudi (2010)

21. Moore, G. M.: Login and set theory. In: Grattan-Guinness, I. (ed.) Companion Encyclopedia of the History and Philosophy of the Mathematical Sciences, pp. 635–643. Roudtledge, London and NY (1992)

22. Moran, J. F., Williams K.: Una Classificazione delle pavimentazioni geometriche realizzate dai Cosmati. Boll. Mat. Ital. Sez. A, ser. VIII, VII-A, 17–47 (April 2004)

23. Mosco, U.: Energy functionals on certain fractal structures. J. Convex Anal. 9(2), 581–600 (2002)

24. Ott, E.: Chaos in Dynamical Systems, 2nd edn. Cambridge University Press, Cambridge (2002)

25. Ott, E.: Attractor dimensions. http://www.scholarpedia.org/article/Attractor_dimensions, last accessed 2018/04/18

26. Shang, J., et al.: Assembling molecular Sierpiński triangle fractals. Nat. Chem. 7, 389–393 (2015)

27. Sierpinski, W.: Sur une courbe dont tout point est un point de ramification. Compt. Rendus Acad. Sci. Paris 160 (1915), 302–305. Source gallica.bnf.fr/BnF

28. Stewart, I.: Four encounters with Sierpinski’s gasket. Math. Intell. 17(1), 52–64 (1995)

29. Tait, L.: Surface chemistry: self-assembling Sierpiński triangles. Nat. Chem. 7, 370–371 (2015)

30. Williams, K.: The pavements of the cosmati. Math. Intell. 19(1), 41–45 (Winter 1997)

31. “Common Mistakes” at https://users.math.yale.edu/public_html/People/frame/Fractals/Panorama/Welcome.html, last accessed 2018/04/30

## Acknowledgements

We would like to acknowledge Corrado Falcolini for useful conversations along these lines.

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Correspondence to Laura Tedeschini Lalli .

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### Cite this paper

Brunori, P., Magrone, P., Lalli, L.T. (2019). Imperial Porphiry and Golden Leaf: Sierpinski Triangle in a Medieval Roman Cloister. In: Cocchiarella, L. (eds) ICGG 2018 - Proceedings of the 18th International Conference on Geometry and Graphics. ICGG 2018. Advances in Intelligent Systems and Computing, vol 809. Springer, Cham. https://doi.org/10.1007/978-3-319-95588-9_49

• DOI: https://doi.org/10.1007/978-3-319-95588-9_49

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