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

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


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.


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

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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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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.

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