Multiplying and Dividing Irrationally

This chapter describes how to multiply and divide, albeit approximately, by some of the world’s most famous irrational numbers, such as π‎, Euler’s number e, 2, 3, both of which occur frequently in the study of triangles, and the Golden Ratio, also sometimes called the Divine proportion. The approximations for π‎ stem from the Ancient World, including the Hebrew Bible, Greek, and Babylonian approximations. An example for 2 is provided by the medieval Jewish polymath, Maimonides. By use of various approximations, the sine of some angles can be easily computed, which can impress those with a grasp of elementary trigonometry. Some examples of “almost” formulas are presented.

The Image of an Architect and Masonic Symbols in Works by Milorad Pavić

The paper analyzes works by the Serbian postmodernist writer Milorad Pavić. It attempts to prove that he possesses knowledge of royal art and uses masonic symbols in his writing related to geometry and architecture, including the radiant delta, compass, masonic gloves, and clepsydra. It is assumed that under the influence of these particular ideas, the writer creates the leading image of an architect and the motif of construction as freemasons believe in the Great Architect of the Universe. In the short novel Damascene, according to speculative masonry’s beliefs, the building of the church projects the building of a temple in a human soul. M. Pavić, as an architect, creates a structure of every novel, which he identifies with the golden section. This paper finds special symbols of the divine proportion in his prose, including snail’s shells, pyramids, and violins. A dynamic structure as an embodiment of the open work concept and a broad spectrum of themes provide artistic communication with a creative recipient. A reader has an opportunity to choose their own style of reading and solving textual puzzles because Pavić’s prose represents a wide variety of themes, symbols, images, and allusions that embody the secrets of Freemasonry, allowing for various interpretations.

The Golden Number

The golden number or divine proportion was defined by Euclid. It is sometimes claimed that it was used in classical architecture, but it is not mentioned by Vitruvius, so this seems unlikely. The illustrations for Luca Pacioli’s book The Divine Proportion were drawn by Leonardo. The golden number is related to the structure of polyhedra with five-fold symmetry. Chapter 13 considers some of the properties of the regular and semi-regular or Archimedean polyhedra, and also considers the suggestion that the pupil in the famous painting of Luca Pacioli is a young Albrecht Dürer.

Golden Section and Golden Rectangles When Building Icosahedron, Dodecahedron and Archimedean Solids Based On Them

A brief history of the development of the regular polyhedrons theory is given. The work introduces the reader to modelling of the two most complex regular polyhedrons – Platonic solids: icosahedron and dodecahedron, in AutoCAD package. It is suggested to apply the method of the icosahedron and dodecahedron building using rectangles with their sides’ ratio like in the golden section, having taken the icosahedron’s golden rectangles as a basis. This method is well-known-of and is used for icosahedron, but is extremely rarely applied to dodecahedron, as in the available literature it is suggested to build the latter one as a figure dual to icosahedron. The work provides information on the first mentioning of this building method by an Italian mathematician L. Pacioli in his Divine Proportion book. In 1937, a Soviet mathematician D.I. Perepelkin published a paper On One Building Case of the Regular Icosahedron and Regular Dodecahedron, where he noted that this “method is not very well known of” and provided a building based “solely on dividing an intercept in the golden section ratio”. Taking into account the simplicity and good visualization of the building based on golden rectangles, a computer modeling of icosahedron and dodecahedron inscribed in a regular hexahedron is performed in the article. Given that, if we think in terms of the golden section concepts, the bigger side of the rectangle equals a whole intercept – side of the regular hexahedron, and the smaller sides of the icosahedron and dodecahedron rectangles are calculated as parts of the golden section ratio (of the bigger part and the smaller one, respectively). It is demonstrated how, using the scheme of a wireframe image of the dual connection of these polyhedrons as a basis, to calculate the sides of the rectangles in the golden section ratio in order to build an “infinite” cascade of these dual figures, as well as to build the icosahedron and dodecahedron circumscribed about the regular hexahedron. The method based on using the golden-section rectangles is also applied to building semiregular polyhedrons – Archimedean solids: a truncated icosahedron, truncated dodecahedron, icosidodecahedron, rhombicosidodecahedron, and rhombitruncated icosidodecahedron, which are based on icosahedron and dodecahedron.

Why does Aristotle think bees are divine? Proportion, triplicity and order in the natural world

Concluding his discussion of bee reproduction in Book 3 ofGeneration of Animals, Aristotle makes a famous methodological pronouncement about the relationship between sense perception and theory in natural history. In the very next sentence, he casually remarks that the unique method of reproduction that he finds in bees should not be surprising, since bees have something ‘divine’ about them. Although the methodological pronouncement gets a fair bit of scholarly attention, and although Aristotle's theological commitments in cosmology and metaphysics are well known, scholars have almost universally passed over the comment about bees and divinity in silence. This paper aims to show why that comment is no mere throwaway, and offers an exploration and elaboration of the ways in which divinity operates even at fairly mundane levels in his natural philosophy, as an important Aristotelian explanation for order, proportion and rationality, even in the lowest of animals.