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Euclid Euclidian Geometry Euclidian geometry, the rules of which are listed in this resource folder, forms the basis for many of Peter Randall-Page’s works. ‘Seed’ (2007), the massive granite sculpture installed in The Clore Education building at the Eden Project in Cornwall, provides a good example of how Peter works. The overall egg-shape of ‘Seed’ results from his sense of the instinctive “rightness” of its contours in relation to size, mass and material. A set of nodes, systematically graded in their diameter and protrusion, spiral around the form. He initially worked with a computer expert, intending to project a mathematically generated configuration onto the surface of the stone. However, the method proved not to deliver what was needed and he resorted to the traditional Euclidian tools of straight- edge and compass. As Peter explains: I plotted the pattern of around 1,800 nodes directly onto the surface of the stone using a ruler and compass. I drew two primary spirals traversing the form in opposite directions to represent the two dominant alignments of circles. Using horizontal bands I was then able to divide each circumference into numerical divisions of two consecutive Fibonacci numbers [a sequence of numbers in which each number is a sum of the previous two], in this case 21 and 34. Joining these points created two families of opposing spirals whose intersection represented the centre of each node. Increasingly in Peter’s recent works the overall shapes and some of the processes have moved in a more irregular direction, from mathematical formulae to organic unpredictability. He has started with a series of found boulders, for example ‘Exhalation’ (2008) and ‘Inhalation’ (2008) which themselves have been shaped by their exposure to the elements, rendering each unique in shape. The overall forms are therefore somewhat different from the shells, pine cones and sunflower seeds that characterise his earlier work. Exhalation (2008) Inhalation (2008) Seed (2007) Euclid Euclid (Greek: Εὐκλείδης — Eukleídēs), fl. 300 BCE, also known as Euclid of Alexandria, was a Greek mathematician and is often referred to as the Father of Geometry. He was active in Alexandria during the reign of Ptolemy I (323 BCE – 283 BCE). His work ‘Elements’ is the most successful textbook in the history of mathematics. In it, the principles of what is now called Euclidean geometry were worked out from a small set of sayings. Euclid also wrote works on perspective, conic sections, spherical geometry, number theory and rigor. Biography Little is known about Euclid except his writings. An Arabian author, al- Qifti (d. 1248), recorded that Euclid's father was Naucrates and his grandfather was Zenarchus, that he was a Greek, born in Tyre and lived in Damascus. But there is no real proof that this is the same Euclid. In fact, another man, Euclid of Megara, a philosopher who lived at the time of Plato, is often confused with Euclid of Alexandria. Of Euclid’s life nothing is known except what the Greek philosopher Proclus (c. CE 410–485) reports in his “summary” of famous Greek mathematicians. According to him, Euclid taught at Alexandria in the time of Ptolemy I Soter, who reigned over Egypt from 323 to 285 BCE. Medieval translators and editors often confused him with the philosopher Eukleides of Megara, a contemporary of Plato about a century before, and therefore called him Megarensis. Proclus supported his date for Euclid by writing “Ptolemy once asked Euclid if there was not a shorter road to geometry than through the ‘Elements’, and Euclid replied that there was “no royal road to geometry”. Today few historians challenge the consensus that Euclid was older than Archimedes (c. 290/280–212/211 BCE). Elements This image shows one of the oldest surviving fragments of Euclid's ‘Elements’, found at Oxyrhynchus and dated to circa CE 100. The diagram accompanies Book II, Proposition 5. Although many of the results in ‘Elements’ originated with earlier mathematicians, one of Euclid's accomplishments was to present them in a single, logically coherent framework, making it easy to use and easy to reference, including a system of rigorous mathematical proofs that remains the basis of mathematics 23 centuries later. The ‘Elements’ is divided into thirteen books which cover plane geometry, arithmetic and number theory, irrational numbers, and solid geometry. Euclid organized the known geometrical ideas, starting with simple definitions, axioms, formed statements called theorems, and set forth methods for logical proofs. He began with accepted mathematical truths, axioms and postulates, and demonstrated logically 467 propositions in plane and solid geometry. One of the proofs was for the theorem of Pythagoras, proving that the equation is always true for every right triangle. The Elements was the most widely used textbook of all time, has appeared in more than 1,000 editions since printing was invented, was still found in classrooms until the twentieth century, and is thought to have sold more copies than any book other than the Bible. Euclid used an approach called the "synthetic approach" to present his theorems. Using this method, one progresses in a series of logical steps from the known to the unknown. Euclid proved that it is impossible to find the "largest prime number," because if you take the largest known prime number, add 1 to the product of all the primes up to and including it, you will get another prime number. Euclid's proof for this theorem is generally accepted as one of the "classic" proofs because of its conciseness and clarity. Millions of prime numbers are known to exist, and more are being added by mathematicians and computer scientists. Mathematicians since Euclid have attempted without success to find a pattern to the sequence of prime numbers. The geometrical system described in the ‘Elements’ was long known simply as ‘geometry’, and was considered to be the only geometry possible. Today, however, that system is often referred to as ‘Euclidean geometry’ to distinguish it from other so-called ‘Non-Euclidean geometries’ that mathematicians discovered in the 19th century. Euclid compiled his ‘Elements’ from a number of works of earlier men. Among these are Hippocrates of Chios (flourished c.460 BCE), not to be confused with the physician Hippocrates of Cos (c. 460– 377 BCE). The latest compiler before Euclid was Theudius, whose textbook was used in the Academy and was probably the one used by Aristotle (384–322 BCE). The older elements were at once superseded by Euclid’s and then forgotten. For his subject matter Euclid doubtless drew upon all his predecessors, but it is clear that the whole design of his work was his own, culminating in the construction of the five regular solids, now known as the Platonic solids. Axioms are statements that are accepted as true. Euclid believed that we can't be sure of any axioms without proof, so he devised logical steps to prove them. Euclid divided his ten axioms, which he called "postulates," into two groups of five. The first five were "Common Notions," because they were common to all sciences: 1 Given two points there is one straight line that joins them. 2 A straight line segment can be prolonged indefinitely. 3 A circle can be constructed when a point for its centre and a distance for its radius are given. 4 All right angles are equal. 5 If a straight line falling on two straight lines makes the interior angles on the same side less than two right angles, the two straight lines, if produced indefinitely, meet on that side on which the angles are less than the two right angles. 6 Things equal to the same thing are equal. 7 If equals are added to equals, the wholes are equal. 8 If equals are subtracted from equals, the remainders are equal. 9 Things that coincide with one another are equal. 10 The whole is greater than a part. The subject of Book II has been called geometric algebra because it states algebraic identities as theorems about equivalent geometric figures. Book III deals with properties of circles and Book IV with the construction of regular polygons, in particular the pentagon. Book V shifts from plane geometry to expound a general theory of ratios and proportions that is attributed by Proclus (along with Book XII) to Eudoxus of Cnidus (c. 390–350 BCE). Book VI applies this theory of ratios to plane geometry, mainly triangles and parallelograms, culminating in the “application of areas,” a procedure for solving quadratic problems by geometric means. Books VII–IX contain elements of number theory, where number (‘arithmos’) means positive integers greater than 1. Beginning with 22 new definitions—such as unity, even, odd, and prime—these books develop various properties of the positive integers. For instance, Book VII describes a method, ‘antanaresis’ (now known as the Euclidean algorithm), for finding the greatest common divisor of two or more numbers; Book VIII examines numbers in continued proportions, now known as geometric sequences (such as ax, ax2, ax3, ax4…); and Book IX proves that there are an infinite number of primes. According to Proclus, Books X and XIII incorporate the work of the Pythagorean Theaetetus (c. 417– 369 BCE). Books XI–XIII examine three-dimensional figures, in Greek ‘stereometria’. Book XI concerns the intersections of planes, lines, and parallelepipeds (solids with parallel parallelograms as opposite faces). Book XII applies Eudoxus’ method of exhaustion to prove that the areas of circles are to one another as the squares of their diameters and that the volumes of spheres are to one another as the cubes of their diameters. Book XIII culminates with the construction of the five regular Platonic solids (pyramid, cube, octahedron, dodecahedron, icosahedron) in a given sphere, as displayed in the image below. The unevenness of the several books and the varied mathematical levels may give the impression that Euclid was but an editor of treatises written by other mathematicians. To some extent this is certainly true, although it is probably impossible to work out which parts are his own and which were adaptations from his predecessors. Euclid’s contemporaries considered his work final and authoritative; if more was to be said, it had to be as commentaries on the ‘Elements’. Almost from the time of its writing, the ‘Elements’ exerted a continuous and major influence on human affairs. It was the primary source of geometric reasoning, theorems, and methods at least until the advent of non-Euclidean geometry in the 19th century. It is sometimes said that, other than the Bible, the ‘Elements’ is the most translated, published, and studied of all the books produced in the Western world. Euclid may not have been a first-class mathematician, but he set a standard for deductive reasoning and geometric instruction that persisted, practically unchanged, for more than 2,000 years. After the Elements Campanus translated The Elements from Arabic to Latin, and the first printed edition appeared in Venice in 1482. The first English translation of The Elements was by the mathematician John Dee in 1570. Dee's lectures and writings revived interest in mathematics in England. His translation was from a Latin translation of an Arabic translation of the original Greek. In 1733, an Italian mathematician named Girolamo Saccheri almost discovered non-Euclidean geometry. He had studied for years in a futile attempt to find a single error in Euclid's postulates. On the verge of a breakthrough, he gave up and published Euclid Cleared of Every Flaw. It would be about a hundred years before another viable geometry was invented. In 1899, the German mathematician David Hilbert presented Foundations of Geometry, the first complete set of geometry axioms since Euclid. Euclid also wrote Data, which contains 94 propositions, Phaenomena, concerning spherical astronomy,Caloptrics, about mirrors, Optics, the theory of perspective, and a work of music theory. In his works about optics, Euclid made light rays part of geometry, working with them as if they were straight lines. Many of the works ascribed to Euclid are no longer in existence or are incomplete. Euclidean constructions The Greek number system made it hard to do calculations. Like the Roman number system, it was not a positional system, had no zero, and had only whole numbers. To compensate for this, they used graphical techniques using a compass and straightedge to produce geometric constructions. These became known as Euclidean Constructions and are described further in Euclidean Constructions - Tools and Rules Other works This detail from ‘The School of Athens’ portrays Euclid, as imagined by Raphael. No likeness or description of Euclid's physical appearance made during his lifetime survived antiquity. Therefore, Euclid's depiction in works of art depends on the artist's imagination. In addition to the ‘Elements’, at least five works of Euclid have survived to the present day. They follow the same logical structure as ‘Elements’, with definitions and proved propositions. ‘Data’ deals with the nature and implications of "given" information in geometrical problems; the subject matter is closely related to the first four books of the Elements. ‘On Divisions of Figures’, which survives only partially in Arabic translation, concerns the division of geometrical figures into two or more equal parts or into parts in given ratios. It is similar to a third century CE work by Heron of Alexandria. ‘Catoptrics’ concerns the mathematical theory of mirrors; particularly the images formed in plane and spherical concave mirrors. The attribution to Euclid is doubtful. Its author may have been Theon of Alexandria. ‘Phenomena’, a treatise on spherical astronomy, survives in Greek; it is quite similar to ‘On the Moving Sphere’ by Autolycus of Pitane, who flourished around 310 BCE. ‘Optics’ is the earliest surviving Greek treatise on perspective. In its definitions Euclid follows the Platonic tradition that vision is caused by discrete rays which emanate from the eye. One important definition is the fourth: "Things seen under a greater angle appear greater, and those under a lesser angle less, while those under equal angles appear equal." In the 36 propositions that follow, Euclid relates the apparent size of an object to its distance from the eye and investigates the apparent shapes of cylinders and cones when viewed from different angles. Proposition 45 is interesting, proving that for any two unequal magnitudes, there is a point from which the two appear equal. Pappus believed these results to be important in astronomy and included Euclid's ‘Optics’, along with his ‘Phaenomena’, in the ‘Little Astronomy’, a compendium of smaller works to be studied before the ‘Syntaxis’ (Almagest) of Claudius Ptolemy. Other works are credibly attributed to Euclid, but have been lost: ‘Conics’ was a work on conic sections that was later extended by Apollonius of Perga into his famous work on the subject. It is likely that the first four books of Apollonius' work come directly from Euclid. According to Pappus, "Apollonius, having completed Euclid's four books of conics and added four others, handed down eight volumes of conics." The Conics of Apollonius quickly supplanted the former work, and by the time of Pappus, Euclid's work was already lost. ‘Porisms’ might have been an outgrowth of Euclid's work with conic sections, but the exact meaning of the title is controversial. ‘Pseudaria’, or ‘Book of Fallacies’, was an elementary text about errors in reasoning. ‘Surface Loci’ concerned either loci (sets of points) on surfaces or loci which were themselves surfaces; under the latter interpretation, it has been hypothesized that the work might have dealt with quadric surfaces. Several works on mechanics are attributed to Euclid by Arabic sources. ‘On the Heavy and the Light’ contains, in nine definitions and five propositions, Aristotelian notions of moving bodies and the concept of specific gravity. ‘On the Balance’ treats the theory of the lever in a similarly Euclidean manner, containing one definition, two axioms, and four propositions. A third fragment, on the circles described by the ends of a moving lever, contains four propositions. These three works complement each other in such a way that it has been suggested that they are remnants of a single treatise on mechanics written by Euclid. Euclid’s orchard In mathematics Euclid's orchard is an array of one-dimensional trees of unit height planted at the lattice points in one quadrant of a square lattice. More formally, Euclid's orchard is the set of line segments from (i, j, 0) to (i, j, 1) where i and j are positive integers. Plan view of one corner of Euclid's orchard. Trees marked by a solid blue dot are visible from the origin. Perspective view of Euclid's orchard from the origin. The trees visible from the origin are those at lattice points (m, n, 0) where m and n are coprime, i.e., where the fraction m⁄n is in reduced form. The name ‘Euclid's orchard’ is derived from the Euclidean algorithm. If the orchard is projected relative to the origin onto the plane x+y=1 (or, equivalently, drawn in perspective from a viewpoint at the origin) the tops of the trees form a graph of Thomae's function. The point (m, n, 1) projects to Euclid's orchard is mentioned in Brutus, the 10th episode of season 3 of ‘Numb3rs’. Abraham Lincoln At 40, Abraham Lincoln studied Euclid for training in reasoning, and as a travelling lawyer on horseback, kept a copy of Euclid's ‘Elements’ in his saddlebag. In his biography of Lincoln, his law partner Billy Herndon tells how late at night Lincoln would lie on the floor studying Euclid's geometry by lamplight. Lincoln's logical speeches and some of his phrases such as "dedicated to the proposition" in the Gettysburg address are attributed to his reading of Euclid. Lincoln explains why he was motivated to read Euclid: "In the course of my law reading I constantly came upon the word “demonstrate". I thought at first that I understood its meaning, but soon became satisfied that I did not. I said to myself, What do I do when I demonstrate more than when I reason or prove? How does demonstration differ from any other proof? I consulted Webster's Dictionary. They told of 'certain proof,' 'proof beyond the possibility of doubt'; but I could form no idea of what sort of proof that was. I thought a great many things were proved beyond the possibility of doubt, without recourse to any such extraordinary process of reasoning as I understood demonstration to be. I consulted all the dictionaries and books of reference I could find, but with no better results. You might as well have defined blue to a blind man. At last I said,- Lincoln, you never can make a lawyer if you do not understand what demonstrate means; and I left my situation in Springfield, went home to my father's house, and stayed there till I could give any proposition in the six books of Euclid at sight. I then found out what demonstrate means, and went back to my law studies."