The School of Athens, Raphael, Geometry, Euclid, Summer School, Geometry Summer School

Geometry Summer School 2022




26 August 2022-
29 August 2022




The ancient Greek geometer Euclid composed The Elements, a synthesis of Hellenistic geometric science, around the beginning of the second century BCE. It has remained one of the most widely-read and influential works in the western canon ever since, as the unparalleled introduction to the method of geometric construction and proof.

We will explore the influence and impact of The Elements on architecture and the visual arts, philosophy and metaphysics, across several continents and a timespan stretching from late antiquity until the present day in the fourth annual Geometry Summer School. We will consider Euclid’s historical impact, not only on geometry as a design tool, but on philosophers and esoteric thinkers, and how this influence appears in writings, art, and architecture.

The Summer School will engage practically with the contemporary relevance of the timeless geometric principles expounded by Euclid in the context of artistic design and craft application. Through historical manuscripts, we will trace Euclid’s appearance in the Islamic world and medieval Europe, following the constructions and discussions recorded therein. We will make note of Euclidean propositions that emerge in the context of a craft workshop, and address the concerns of theoretical precision and craft approximation in drawing complex geometric designs. As part of the continuing tradition of the Geometry Summer School, we will also explore how a deep understanding of universal principles inspires new avenues of discovery.


Definitions, Propositions, Theorems & Constructions: A introduction to Euclid’s Elements

Presenter: Sarah Gelsinger Brewer

This session of the Geometry Summer School will be a practical review of the basic Euclidean geometric principles that are the driving force behind the compass and straightedge construction of geometric patterns. By working through the very same constructive drawing demonstrations used by Euclid, we will recall the possibilities and limits of our tools and gain confidence in the accuracy of our constructions. The skills reviewed in this session will set the stage for drawing exercises throughout the Summer School.

Tracing Euclid in Medieval Islamic Manuscripts on Geometry” Part I and Part II

Presenter: Katya Nosyreva

Most medieval Islamic authors who wrote on geometry were trained in the method of Euclid’s Elements, which was available in several good Arabic translations. These authors studied geometry in order to deepen their knowledge in mathematics and astronomy, pointing to the interrelatedness of these disciplines. The polymath Abū al-Wafāʾ Būzjānī (940-998) was no exception, and his book entitled Kitāb fī mā yaḥtaj ilayh al-ṣāniʿ min al-aʿmāl al-handasiyya (A Book on Those Geometric Constructions Which Are Necessary for a Craftsman) is our only known geometric work in the Islamic world specifically addressing the needs of craftsmen and artisans.

In his practical manual on geometry, Abū al-Wafāʾ addresses practitioners in various fields of design which required understanding of the methods of geometric constructions. But in proposing different solutions to given problems he was also demonstrating his knowledge as a brilliant theoretical geometrician.

Over two sessions, we will engage in dialogue and drawing based on Būzjānī’s text, tracing his general exercises – potentially useful in a workshop environment – to Euclid’s propositions, and look at his discussion of tools, where he adds a set square to Euclid's ruler and compass, as the three basic instruments necessary for geometric constructions. We will compare those to other extant works in this genre which discuss same constructions and describe additional tools, such as plummet, T-square, movable ruler, and perfect compass, needed for angle trisection or dealing with the science of conics, which fall outside Euclid’s Elements.

We will then move on to Abū al-Wafāʾ discussion of the differing methods of craftsmen and geometricians, and how the approximation-methods of the former are contrasted with the more accurate methods of the latter. We will reconstruct a drawing by Abū al-Wafāʾ based on Euclid’s proposition dealing with congruent areas of squares and rectangles, and see how it developed into a popular motif in Islamic architecture, known as ‘whirling kites’.

Euclid in Arabic (and Persian): A Guided Tour through Digitised Manuscripts

Presenter: Bink Hallum

This session will outline the Islamicate tradition of Euclid’s Elements. Beginning with the earliest translations of Euclidean texts from Greek into Arabic in the 9th century, the development of this tradition will be traced from the early reception of Euclid by the geometers of the 10th and 11th centuries, through the standardisation of the Arabic text of the Elements in the 13th century, to the later commentaries used in school settings. The presentation will be illustrated entirely with digitised manuscripts that are freely available on the web, allowing participants to return to them for further exploration and inspiration at their leisure. With this aim in mind, Dr Hallum will use his curatorial experience to introduce participants to major online Islamic manuscript collections and give practical demonstrations of finding and using manuscripts of interest on the various digital library portals.

Euclid at the workbench

Presenter: Joachim Tantau

In this session Joachim will explore the Euclidian postulates from a maker’s point of view. Joachim is a practicing artist and cabinetmaker and uses the Euclidian tools, compass and straight edge, in all of his designs. Through a series of drawing exercises he will shed a light on the practical implications of Euclidian Geometry and in particular explore ‘workarounds’ and approximations for the impossible constructions like trisecting an angle and squaring a circle as they have been used by artists and craftsmen for centuries.

Exploring Proclus’ Commentary on Book One of Euclid’s Elements

Presenter: Earl Fontainelle

The philosopher Proclus the Successor (c. 410-485 CE) was an immensely influential thinker of late antiquity, teaching a polytheist, religious Platonism at Athens in a Roman empire undergoing catastrophic changes during his lifetime. His work represents the last and greatest synthesis of a tradition, going back to Plato, of creative re-interpretation of Pythagorean and Neopythagorean ideas about mathematics, exploring both the exoteric and esoteric sides of numerical science. Proclus wrote an enormous number of commentaries, the best known of which are comprehensive exegeses of Platonic dialogues like the Timæus and Republic. His works also contain a number of less-well-known writings, among which is the Commentary on Book One of Euclid’s Elements. Proclus considers Euclid’s Elements as far more than merely an instructional, mathematical text.

In his presentation, Dr Fontainelle will explore Proclus’ Commentary, using it as a springboard for exploration of a vast ancient tradition of esoteric mathematical thinking. We will introduce and discuss the historical realities (and fables) of the Pythagorean tradition. We will also explore how Proclus conceptualises geometry, mathematics, and number not solely in terms of mathematical operations, but in terms of Platonist ontology, arithmological speculation, and even the history of the kosmos. Finally, we will consider the role played by geometric and mathematical education in the harmonisation of the soul in late Platonist thought. Proclus’ Commentary on Euclid shows us how esoteric and spiritual approaches to number can be applied to a straightforward mathematical text, drawing forth meanings and correspondences.

Trisecting an angle using origami

Presenter: Ricardo Hinojosa

In this session we will focus our attention on paper folding moves developed to solve the angle trisection problem.

Trisecting an angle is technically impossible to do precisely only using the classic euclidean method of construction: compass and straight-edge.

Although many different clever techniques exist to approach it or to work around a solution by adding other tools, within origami (paper-folding) is all we need. The objective of these exercises is to grasp paper folding as a powerful geometrical tool, learn of the trajectory different mathematicians have reasoned and give them a try.

The beauty is that the mathematics necessary to prove how these processes result in true trisections of arbitrary angles is a challenge in and of itself. Folding the steps leads to a practical end and the underlying truths are automatically satisfied. It takes additional effort to uncover why folding works and that is enlightening.

This is one of the most interesting phenomena of origami: one may perform operations which imply profound geometrical applications all the while being a craft of imaginative heart.

Check back soon for information on additional sessions!


Sarah Brewer

Sarah Gelsinger Brewer is an artist-mathematician from Mobile, Alabama, USA. She taught high school and university-level mathematics for 13 years, during which time she developed elective courses in Advanced Euclidean Geometry, Visual Mathematics, and Geometric Design. She holds a B.F.A. in art history and ceramics, and B.S. and M.S. degrees in pure mathematics with a thesis in the field of topology. She is interested in the intersection of mathematics and the arts, including applications such as DNA topology, origami, wallpaper groups, fractals, mathematical perspective, and data visualization. Her recent area of mathematical and artistic study has been on variable-angled star rosette patterns based on circle packings with k-uniform tiling vertex graphs.

Lisa DeLong

Dr Lisa DeLong completed her doctoral studies at the School in 2007 investigating the principles of geometric design in Islamic and Western traditions. She is the author of Curves: Flowers, Foliates and Flourishes in the Formal Decorative Arts. An avid painter, Lisa is also Outreach Programme Manager for the School, designing and conducting educational workshops internationally."

Earl Fontainelle

Earl Fontainelle is a writer and researcher focusing on the history of western ideas across the longue durée. He specialises in the intersections of philosophy, religion, science, and magic in late antiquity. He directs the Network for the Study of Esotericism in Antiquity and produces the ongoing academic outreach project, The Secret History of Western Esotericism Podcast, a long-form investigation of forgotten and rejected ideas.

Bink Hallum

Bink Hallum is Arabic Scientific Manuscript Curator at the British Library where he scopes and catalogues manuscripts for inclusion in the Qatar Digital Library. He is also Research Fellow in the Department of Classics and Ancient History at the University of Warwick where he is editing and translating a tenth-century Arabic encyclopaedia of alchemy. His doctoral studies (Warburg Institute, 2008) focused on the transmission of Greek alchemical literature into Arabic. He works in the field of the history of ideas, with special interests in the history of science, medicine and magic, transmission of knowledge, historical scholarly networks, philology, Islamic codicology and palaeography.

Ricardo Hinojosa

Ricardo Andrés Hinojosa (also known as Kamikyodai) is a renowned paper folder and origami artist from México. He was born in 1993 and in 2003 was introduced to paper folding. This pastime developed into a passion and since 2014 Ricardo has been teaching people of all ages to fold. In 2016, Ricardo started Kamikyodai (“paper kin” in Japanese) as a home business that offered paper-folded products/art, workshops and folding for hire. From 2015 to 2017, he worked summers making paper in Papel Oaxaca, an artisan paper studio. He then developed origami techniques that serve as drawing tools for geometric Islamic Art. Ricardo has gone on to teach courses with Art of Islamic Pattern and on his own to Muslim communities in the United States and Europe.

Katya Nosyreva

Katya Nosyreva, PhD, is a ceramicist, visual artist, and geometer. Living with her family on Dartmoor, UK, she works with porcelain clay and the visual and symbolic language of sacred geometry. Katya’s PhD research (Prince's School of Traditional Arts, 2013) combined her studio practice with geometry. She designed and made an architectural space for a Sufi centre in Delhi, India. This work explored the practice of traditional craft within contemporary Sufism.

During the course of her PhD research, Katya worked with historical manuals and architectural scrolls on practical geometry. These treasures from the medieval Islamic guild-tradition document practical geometric methods and reveal much about historic approaches. In working with these manuals, Dr Nosyreva finds inspiration for the contemporary geometer.

Joachim Tantau

Joachim Tantau is a cabinetmaker and designer. He graduated from The Prince’s Foundation School of Traditional Arts in 2013 where he received the Barakat Prize for his research into the geometry and design principles of Andalusian Muqarnas.

He currently runs a studio in Hamburg, Germany where he produces bespoke furniture and objects d’art. Joachim also teaches workshops on marquetry and woodworking techniques in the UK, Egypt, Spain and Azerbaijan.

How do I participate?

Sessions will be taught live via Zoom; all sessions will be recorded, meaning you can join us from wherever you are in the world!

The Summer School will be supported by an online learning environment, Thinkific, where students can watch event recordings, explore additional materials and interact with experts and fellow participants.

What materials do I need?

Materials will be specified well in advance of the start date.

Do you offer concessions?

Full-time students and OAPS can apply for a 15% discount.

MA/MPhil/PhD graduates of The Prince's Foundation School of Traditional Arts are also eligible for concessions.

Concessions cannot be applied in retrospect. To receive a concession, please apply via our online form. We will then send you a discount code to use when booking your space.

What if I need to change or cancel my booking?

No refunds, unless cancelled by the School. If you cancel up to 1 week before the workshop starts, we can offer a transfer to another workshop of equivalent value, subject to availability.

Can I buy recordings instead of attending?

No. We encourage you to attend the course in real-time to really benefit from instruction. Students on the course can watch recordings of each session on our online learning platform for a limited time.

Recordings cannot be purchased separately if the course is fully booked or if the course has already started.


You must download the Zoom app and create a free account before the class begins, so that you can use the full range of features:

  • You can create your free account here

  • You can download the Zoom app onto your computer or device here

Tutor Biographies