Anamorphic, or distorted, images problematized the idea that viewing was a neutral act. Anamorphic images in print frequently took portraiture and topographic renderings as their subject matter, the same content around which visual acuity was being developed in printed books.

Educationally, we are in an exciting time in terms of geometrical investigations in the classroom. While the manipulation of concrete materials to enable student construction of two-dimensional figures and three-dimensional objects has been readily available for many years, there are a growing number of mathematics classrooms that have access to dynamic geometry software and interactive sites that enable real-time creation and exploration of geometric figures and their properties. In fact, in some pockets of society, students’ access to a mobile device is in a similar manner to how classrooms of the 1980s used pen and paper as a resource. While, in jest, mobile devices may be referred to ‘an extension of the brain’, in its regular use as an instant source of information and exploration there is an element of this use that can be exploited for positive gain in the mathematics classroom. This chapter explores the development of geometrical concepts and the manner in which we can facilitate exploratory experiences to assist students in their development.

Chapter 3 focusses on the temple of Hera at Foce del Sele north of the Greek colony of Poseidonia-Paestum in southern Italy. New archaeometric analysis on the metopes from the Hera sanctuary near the mouth of the river Sele has made it possible to propose a new reconstruction of the oldest Hera temple on the site, which belongs to the first generation of Doric stone temples. The study of the architectural elements confirms the decorative nature of the first Doric friezes. Moreover, by analyzing the mythological subjects on the frieze and comparing them with other early Doric temples in Selinous, Delphi, and Athens, it can be shown that the tendency to choose Panhellenic themes over local traditions is a general feature of early Doric temples. Because of the detachment of the imagery from local traditions, the Doric temple is described as a “non-place” according to the definition of the French anthropologist Marc Augé. Conceiving temples as standardized “non-places” that could be set up in any given local environment was crucial to the agendas of Greek elites, who needed to reorganize agricultural and urban landscapes to regulate population pressure and social tensions – both in the colonies and in homeland Greece.

This chapter introduces, closely following Wabeladio’s explanations, how the logics and phonographic aspects of Mandombe work as an alphabet, explaining the geometrical principles underpinning it. It also contains some personal reflections on the relationship between mathematics and culture.

Chapter 4, ‘Writing It Down: Innovation, Secrecy, and Print’ explains how mining, and subterranean geometry, evolved during the troubled time of the Thirty Years War (1618–1648). It brings together issues related to book history as well as the history of training and teaching practices. Balthasar Rösler (1605–1673) introduced numerous innovations, and his teaching was disseminated by his students among mining regions, in a series of beautifully illustrated and hitherto unstudied manuscripts. The birth of this technical genre is presented in detail, with its evolution and uses within the training system of mining regions. In 1686, Nicolaus Voigtel then published the first practical textbook on the topic. Surprisingly, the craftsmen’s manuscripts weathered the rise of the printed press. I argue that authoring and publishing books failed to supersede the authority of practitioners precisely because their know-how was embedded in a specific technical and cultural setting. Subterranean geometry would stay an underground knowledge for another century, as most innovations arose within this handwritten tradition.

Chapter 1, ‘Of Scholars and Miners’, introduces the discipline of subterranean geometry from the point of view of Renaissance scholars. Early modern humanists were fascinated by the underground world of metal mines. The richness of the geometrical thinking contained in Georgius Agricola’s De re metallica (1556) or Erasmus Reinhold’s On Surveying (1574) is presented. By comparing them with actual productions of contemporary mine surveyors, I further show that these books, despite their lifelike descriptions and illustrations, did not limit themselves to straightforward, faithful depictions of actual practices. Early modern readers were presented with rational reconstructions and pseudo-technical procedures. In spite of a thorough knowledge and a genuine interest for the underground world, scholars mainly used their writings on mines in a patronage context, or to display their interpretation of Euclidean geometry.

Thematically, formally and structurally, Wallace’s writing concerned itself with the infinite, from the antinomies of set theory and the obese Bombardini in The Broom of the System to the featureless horizon of Peoria in The Pale King, by way of the title of Infinite Jest and the brief and not wholly successful exploration of Cantorian mathematics in Everything and More, the idea of the infinite was never far from any of Wallace’s writing. Moreover, the structures of the writing continually reinscribe this obsession with infinity, with none of the novels conforming to a traditional boundaried structure and the collections of short fiction troubling the very concept of order in their use of pagination and enumeration. This chapter illuminates the importance of infinity to Wallace’s writing by exploring its formal and thematic development through his career, demonstrating that infinity worked as a conceptual counterpoint to solipsism, both an existential threat and a source of profound hope for the disassociated subject of contemporary culture.

This paper is detective work. I aim to show that the brilliant Pythagorean mathematician Archytas of Tarentum was the founder of ancient Greek mathematical optics. The evidence is indirect. (1) A fragment of Aristotle preserved in Iamblichus is one of two doxographical notices to mention Pythagorean work in optics. (2) Apuleius credits Archytas with a theory of visual rays which saves the principle that the angle of reflection is equal to the angle of incidence. I argue that the source from which Apuleius got this information was the Catoptrics of Archimedes, the genuineness of which I defend against Knorr’s hypothesis that it is the Euclidean Catoptrics, which had been misattributed to Archimedes. (3) The omission of optics from the mathematical curriculum in Plato’s Republic, and the Timaeus’ wholly physical account of mirror images, can be explained as polemical, for it is well attested that optics was practised in the Academy. The reason Plato does not mention optics is that he objected to Archytas using mathematics to understand the physical world rather than to transcend it.

As an appendix, we can look briefly at the central ideas of General Relativity (though we are limited, since much of the maths is beyond our scope). We prepare the ground with a number of thought experiments, and then discuss, in outline, the geometrical ideas we have to use. We can get a sense of what Einstein's equation is doing, and we look at some solutions of Einstein's equation (including the Schwarzschild metric), describing possible spacetimes.

Chapter 8 focuses on Kant’s reaction to the metaphysics of quantity found in Leibnizian and Wolffian rationalism. Leibniz had broad ambitions for a unified theory of all knowledge that subsumed mathematics under metaphysics. Leibniz accordingly sought metaphysical definitions of quality and quantity that in turn supported metaphysical definitions of similarity and equality as identity of quality and quantity, respectively. A criterion of success was that these definitions corresponded to Euclid’s geometrical notions of similarity and equality. This chapter examines Leibniz, Wolff, and Baumgarten’s views of quality and quantity and the contrast between them, which was closely tied to the conditions of their representation and distinct cognition. Kant adopts some of their understanding of the metaphysics of quantity, such as the definitions of similarity and equality as identity of quality and quantity, respectively. At the same time, he radically reforms it. Kant distinguishes between two notions of quantity, quanta and quantitas, and hence draws two contrasts with two corresponding notions of quality: quality versus quantum, and quality versus quantitas. Most importantly, Kant holds that quanta require intuition for their representation. This preserves the general framework of the Leibnizian and Wolffian metaphysics of quantity while radically reforming it at its foundation.

This chapter expands a little on the idea that gravity is geometry, and then describes how the geometry of space and time is a subject for experiment and theory in physics. In a gravitational field, all bodies with the same initial conditions will follow the same curve in space and time. Einstein’s idea was that this uniqueness of path could be explained in terms of the geometry of the four-dimensional union of space and time called spacetime. Specifically, he proposed that the presence of a mass such as Earth curves the geometry of spacetime nearby, and that, in the absence of any other forces, all bodies move on the straight paths in this curved spacetime. We explore how simple three-dimensional geometries can be thought of as curved surfaces in a hypothetical four-dimensional Euclidean space. The key to a general description of geometry is to use differential and integral calculus to reduce all geometry to a specification of the distance between each pair of nearby points.

To understand ways we might infer stellar distances, we first consider how we intuitively estimate distance in our everyday world, through apparent angular size, and/or using our stereoscopic vision. We explain a practical, quite direct way to infer distances to relatively nearby stars, namely through the method of trigonometric parallax. This leads to the definition of the astronomical unit and parsec, and the concept of solid angles on the sky, measured in steradians or square degrees.