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Visualizing 3D

Visualizing 3D. Between Measurement and Illusion Dan Collins VizProto. Visualization is….

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Visualizing 3D

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  1. Visualizing 3D Between Measurement and Illusion Dan Collins VizProto

  2. Visualization is…. • Visualization is a method of computing. It transforms the symbolic into the geometric, enabling researchers to observe their simulations and computations. Visualization offers a method for seeing the unseen. It enriches the process of scientific discovery and fosters profound and unexpected insights. In many fields it is already revolutionizing the way scientists do science. • SIGGRAPH proceedings, 1987. B. McCormick, T. DeFanti, and M. Brown [MCC87]

  3. Euclid300 B.C. • Euclid's "Elements," written about 300 B.C., a comprehensive treatise on geometry, proportions, and the theory of numbers, is the most long-lived of all mathematical works. This manuscript preserves an early version of the text. Shown here is Book I Proposition 47, the Pythagorean Theorem: the square on the hypotenuse of a right triangle is equal to the sum of the squares on the sides. This is a famous and important theorem that receives many notes in the manuscript. • Pythagorean Theorem

  4. Euclid’s description of the Pythagorean theorem

  5. Euclid300 B.C. • Euclid also wrote Optica,the first text on geometrical optics, in which he defines the terms visual ray and visual cone. • He noted that light travels in straight lines and described the law of reflection. He believed that vision involves rays going from the eyes to the object seen and he studied the relationship between the apparent sizes of objects and the angles in which they meet at the eye. Raphael, The School of Athens,1509, Fresco, Vatican, Rome Detail showing Euclid with his students

  6. Pythagorus580-520 B.C. • Pythagorus was a mathematician who made important contributions to geometry. "He was a Greek philosopher and religious leader who was responsible for important developments in the areas of mathematics, astronomy, and music theory. He was also a healer, a wrestler, and was politically active. He founded a philosophical and religious school which has come to be known as the Pythagorean Society. • The Pythagoreans saw that many things in the universe were related in ways that could be stated in numbers. They reasoned that numbers must be the 'stuff' philosophers were looking for. The universe including man is a closed system. Both can be understood by the relation of the parts. These relations can be expressed in terms of numbers. These ideas led them to believe that if one could penetrate the secrets of numbers, he would penetrate the secrets of the universe and the destiny of man. This led to the careful study of geometry, the highest form of mathematics.”

  7. PantheonRome, Italy, 118 to 126 ADArchitect unknown Exterior view of the Pantheon in modern day Rome Interior view of the Pantheon Giovanni Paolo Panini, c. 1750

  8. The Pantheon and the Neo-Pythagoreans • The Roman pantheon can be considered an architectural image of the Greek Pythagorean cosmos, a "living organism" with a mathematically-proportioning "soul" and unchanging, "eternal" consonant-symphonic ratios. To generate harmony, the laws of arithmetic, geometry, astronomy and musical-proportions are fused. It "resembles the heavens", but is a resemblance based on mathematical knowledge, a summary of the ancient quadrivium*.” • --Girt Sperling • * The quadrivium was the higher division of the seven liberal arts in the Middle Ages, composed of geometry, astronomy, arithmetic, and music. Section showing pythagorean ratios at work in the Pantheon.

  9. Before Perspective • Perspectival errors appear in paintings usually done before 1400. • The perspective lines usually converge, but not to a single point and not on the horizon. • Initial word panel of Psalm from the Kaufmann Haggadah. Spain, late 14th C.

  10. Brunelleschi1377-1446 • Brunelleschi designed the stupendous dome which crowns the cathedral in Florence, a work which occupied him intermittently from 1417 to 1434. The technical difficulties involved in erecting the new dome underscore an important aspect of his talents: he was a daring innovator, with a solid knowledge of math and mechanics.

  11. Brunelleschi • He developed many important construction methods as well as contributing to the evolution of perspective. His mathematical work led to the invention of linear perspective.

  12. Brunelleschi • Filippo Brunelleschi was the first to carry out a series of optical experiments that led to a mathematical theory of perspective.. Brunelleschi used his training as a gold smith to apply a silver background on a painted panel, allowing the color of the sky and passing clouds to become part of the painting as seen by the viewer. This was an attempt at a perspective painting and interactive art. The panel was constructed with a hole at the vanishing point. The reflection of the image was viewed in a mirror through the hole, giving an illusion of depth. • http://library.thinkquest.org/3257/illusion.html#peep

  13. Brunelleschi devised a method of perspective for architectural purposes: he is said by Manetti to have made a ground plan for the Church of Santo Spirito in Florence on the basis of which he produced a perspective drawing to show his clients how it would look after it was built.

  14. Masaccio1401-1428 • Masaccio's Trinity, 1427-28 Santa Maria Novella, Florence (6.67 x 3.17 m) is often used to illustrate the early culmination of mathematical perspective experiments.

  15. Alberti1404 - 1472 • ALBERTI'S WINDOW • The traditional form of pictorial representation using perspective methods developed by Renaissance artists is sometimes referred to as Alberti's Window. • This is because, in his treatise Della pittura, On Painting, 1435-6, the Classical theorist and painter Leon Battista Alberti noted that, when he set out to paint a scene on a panel, he assumed the picture would represent the visible world as if he were looking through a window. Some artists did, in fact, create grids across the opening of a window and transfer the scene to a gridded canvas as compelling evidence that western perspective was a natural form of representation. Alberti’s “fenestre” (window) or “velo”

  16. Alberti • Alberti's Construction System • 1. B-one braccio module (one third of the height of a man). The base of the picture is divided into braccia. The height of the man at the front plane of the picture gives the level of the horizon, H. • 2. The braccio divisions are joined to the perspective focus, V, to give the orthogonals. • 3. In side elevation, lines are drawn from braccio divisions behind the picture plane P to the eye at E. The points of intersection on P are noted. • 4.The levels of the points of intersection are marked at the side of the picture plane, and locate the horizontal divisions of the tiles. Z is the 'distance' point, though Alberti only mentions using one diagonal to check the construction.

  17. Paolo Uccello • Among the best examples of early uses of linear perspective is Paolo Uccello's fresco of the "Deluge" in Florence, completed about 1448. Here linear perspective is used to present an elaborate architectural setting. The real object of fascination, however, is Uccello's rendering of the mazzocchi, the curious checkered hats, of which there are two in "The Deluge.” Ucello had actually drawn such wonderful polyhedral forms in studies of perspective drawings, and these clearly demonstrate the mastery he had of the new mathematical techniques.

  18. Piero della Francescac.1420 - 1492 • The culmination of the mathematical theory of perspective with a philosophical program of the most intense and religious order comes with the work of Piero della Francesca. His St. Anthony's Polyptich, in Perrugia, shows how masterfully he was able to use the new theory of perspective.

  19. Leonardo1452 - 1519 • Perspective is nothing else than the seeing of an object through a sheet of glass, on the surface of which may be marked all the things that are behind the glass --Leonardo da Vinci • Leonardo studied optics from both the scienitific and the artistic points of view. He believed that painting should be considered a Liberal Art because it was based on mathematically derived perspective theory and satisfied the primary sense of sight. Da Vinci realized that unless a person viewed a painting through a peephole, the visual image would be different than the image the artist painted.

  20. Dürer1471-1528 • One of several machines invented by Dürer for making perspectival drawings consisted of a needle driven into the wall and a piece of string and a hinged frame. The piece of string has a pin on one end and a weight on the other; between the eye of the needle and the object is placed a wooden frame within which every point can be determined by two movable threads crossing each other at right angles. When the pin is put on a certain point of the object the place where the string passes through the frame determines the location of that point within the future picture. This point is fixed by adjusting the two movable threads and is at once entered upon a piece of paper hinged to the frame; and by a repetition of this process the whole object may be transferred gradually to the drawing sheet. A woodcut from Albrecht Dürer's treatise on measurement Underweysung der Messung, 1527

  21. Dürer1471-1528 • The Perspectograph is an instrument that allows the user to obtain, point by point, a correct perspective drawing of a three dimensional object. Perspectographs were used by painters and sceno-graphers in 16th and 17th cent. (and as early as the 15th cent. by Alberti). Some types of Perpespectograph are very simple (as these reproduced in Dürer's xylographies), some types are rather complex. In this model of Dürer's perspectograph, an observer looking through the ocular sees the pattern drawn on the vertical table exactly superimposed on the pattern drawn on the horizontal table. Dürer's Perspectograph, early 16th c. (replica)

  22. Palladio1508-1580 • The art historian Rudolph Wittkower writes, "The conviction that architecture is a science, and that each part of a building, inside as well as outside, has to be integrated into one and the same system of mathematical ratios, may be called the basic axiom of Renaissance architects." Many modern authors have analyzed Wittkower's thesis that harmonic proportions derived from musical scales played a central role in the minds and designs of Renaissance theorists and architects. Central to this debate is Palladio's oeuvre--his architecture and his Quattro libri (four books). • --Stephen R. Wassell Elevation and plan of a typical Palladian villa.

  23. vitruviusc. 90-20 B.C.E. • Images by Cesare Cesariano (1521) • This is a profusely illustrated edition of the most famous of antique texts on architecture, The Ten Books on Architecture. It was known throughout the Middle Ages, in multiple copies and probably versions.

  24. Hans Holbein, The Ambassadors, 1536

  25. Francesco Borromini1599 - 1667 This architectural tromp l'oeil of an actual "perspective" collonade in the Palazzo Spada, was fashioned by Galileo's contemporary, Borromini in 1653. This is actually an illusion, played with the help of mathematical perspective. The trick is revealed in the image at the right where two figures of equal height show the perspective at work. The image in the center is a modern CAD rendering.

  26. Diderot1713 - 1784 • Denis Diderot was the creator of the first Encyclopedia in 1751. More then 160 authors contributed to the encyclopedia. By 1789 there were nearly 16,000 copies sold. The pope placed the encyclopedia on the Index of Prohibited books. • In his discussions on art, he provides the broader social context for the arts. In his entries on Art, he describes the origin of the sciences and arts, their distribution into liberal and mechanical arts, the goal of the arts, and his own project for a general treatise on the mechanical arts. • We began by making observations on the nature, service, usage, qualities of beings & of their symbols; then we gave the name of science or of art or of discipline in general, to the center or unifying point to which we related the observations that we had made, to form a system of either rules or instruments, & of rules tending towards the same goal; because that is what a discipline is in general. (ART, in Diderot & d'Alembert, 1751-1772, Vol. 1, p. 713)

  27. Contour plot . Map of Paris by L. L. Vauthier (1874), showing population density by contour lines, the first statistical use of a contour map. This approach to representing multivariate data arose from the use of contour maps in physical geography showing surface elevation (first published in 1752 by Buache), which became common in the early 19th century. It was not until 1843, however, that this idea was applied to data, when Léon Lalanne constructed the first contour plot, showing the mean temperature, by hour of the day and by month at Halle (lower left). Lalanne's data formed a regularly-spaced grid, and it was fairly easy to determine the isolines of constant temperature. Vauthier generalized the idea to three-way data with arbitrary (x,y) values in his map of the population density of Paris. http://www.math.yorku.ca/SCS/Gallery/noframes.html

  28. This figure (showing the population of Sweden from 1750-1875 by age groups) by Luigi Perozzo, from the Annali di Statistica, 1880, is a very early example of a 3D stereogram. Perozzo's figure is also notable for being printed in color in a statistics journal, and in a way which enhances the perception of depth.

  29. Etienne-Jules Marey1830- 1906 Etienne-Jules Marey, 1830-1906, was among the pioneers of dynamic graphics and the graphical representation of movement and dynamic phenomena. This image, from Marey's La méthode graphique dans les sciences experimentales (1876, p. 150) compares the time course of respiration of a person at rest and under exertion, using a pen-recording device to plot the traces over time.

  30. Mapping the London Underground • Harry Beck's 1933 diagram of the 7+ lines of the London Underground, although geographically inaccurate, provides a coherent overview of a complex system. (See map at upper left). • With excellent color printing, classic British railroad typography (by Edward Johnson), and, in the modern style, only horizontal, vertical, and 45 degree lines, the map became a beautiful organizing image of London. • For apparently quite a number of people, the map organized London (rather than London organizing the map). Despite 70 years of revision due to extensions of the Underground and bureaucratic tinkering (the marketing department wrecked the map for several years), the map nicely survives to this day. • Compare map from late 1920s at lower left.

  31. The History of CAD • 25 years ago, nearly every drawing produced in the world was done with pencil or ink on paper. Minor changes meant erasing and redrawing while major changes often meant recreating the drawing from the scratch. If a change to one drawing affected other documents you were dependent upon having someone manually recognize the need to make the changes to the other drawings and to do so. • CAD has fundamentally changed design and the way we “visualize” 3D.

  32. Dynamic Visualization • Visualization of Storm patterns combines 3D graphics and actual metrics

  33. Visualizing the WebGunilla Elam,Warriors of the Net, 1999 • Elam’s background is in fine arts and she also did research into the social aspects of computing and networking technologies at the Ericsson Medialab and now works as a designer at a startup venture called AirClic. Of the many challenges in making Warriors of the Net, Elam says that, “The hardest part was without question to simplify the structure into an understandable, easy to grasp concept. I had not been going into the tech part of the Internet much before starting with this, so the way we did it was Tomas filling me up with as much information I could handle, then let me think about it for a while and melt it down to a level where anyone would be able to understand it.”

  34. Decision Theater at ASU • East Valley Water Forum (EVWF) • The EVWF is a regional cooperative of water providers who are working with Arizona Department of Water Resources (ADWR) with support from the Bureau of Reclamation to develop data driven scenarios about ground water policy issues under a variety of drought scenarios. Their work with the Decision Theater will assist them in developing informed planning decisions as the east portion of the Salt River Valley continues its explosive growth. Key collaborators: K. Sorenson (City of Mesa) and D. Mason (ADWR).

  35. Decision Theater at ASU • Urban Heat Island (UHI) • The UHI explores and models heat retention in the Phoenix metropolitan area. The effect of UHI during Arizona summers has been a 12 degree rise in night time low temperatures in the last 20 years. Scientists have developed predictive models based on dynamic changes in land use that can help planners and decision makers better understand the UHI phenomenon. The goals are to understand probable impacts of UHI on planning urban systems (such as electrical capacity to accommodate increased power use for air conditioning) and to explore the effectiveness and impact of potential solutions for mitigation.

  36. Decision Theater at ASU • Environmental Fluid Dynamics Program • Typically, computational fluid dynamics models of atmospheric events are presented as numeric data or 2 dimensional graphics. Data from a Defense Threat Reduction Agency (DTRA) funded project to simulate anthrax release in Oklahoma City has been modeled and visualized as a 3D animation. This work provides a foundation for developing interactive scenarios to study the effects of wind direction, wind speed, and building design on dissemination of bacterial agents. The research permits informed training of emergency response teams to real natural or man made emergencies.

  37. The History of CAD (pre-1970) • The first graphic system was in mid 1950 the US Air Force's SAGE (Semi Automatic Ground Environment) air defense system. The system was developed at MIT’s Lincoln Laboratory. The system involved the use of CTR displays to show computer-processed radar data and other information. • In 1960, Ivan Sutherland used TX-2 computer produced at MIT's Lincoln Laboratory to produce a project called SKETCHPAD, which is considered the first step to CAD industry. • In 1960 McDonnell Douglas Automation Company (McAuto) founded. It will play a major role on CAD developments. • The first Computer-Aided Design programs used simple algorithms to display patterns of lines at first in two dimensions, and then in 3-D. Early work in this direction had been produced by Prof. Charles Eastman at Carnegie-Mellon University, the Building Description System is a library of several hundred thousands architectural elements, which can be assembled and drawn on screen into a complete design concept. • In mid 1960 large computers characterized the period, vector display terminals and software development done in assembly language. The only significant attempt to create a commercially CAD system was Control Data Corporation's Digigraphics division, a successor to the previously mentioned ITEK. The system costs half million dollars and were sold in few units. • In 1968 Donald Welbourn had the vision to see the possibility of using computers to assist pattern makers to solve the problems of modelling difficult 3D shapes. Today we take for granted 3D modelling, in 1968 only crude 2D drawing systems were available using terminals linked to large main frame computers. • David Evans and Ivan Sutherland founded in 1968 Evans and Sutherland. • In 1969 were founding Computervision and Applicon companies. Computervision was created to produce systems for production drafting and in the same year it sold the first commercial CAD system to Xerox.

  38. The History of CAD (1970-1980) • At the end of 70s a typical CAD system was a 16-bit minicomputer with maximum of 512 Kb memory and 20 to 300 Mb disk storage at a price of $125,000 USD.

  39. The History of CAD (1980-1990) • 1981: Computer graphics from Cornell University founded 3D/Eye Inc., a pioneered 3D and graphics technology.Unigraphics introduced the first solid modeling system, UniSolid. It was based on PADL-2, and was sold as a stand-alone product to Unigraphics. • 1982: CATIA Version 1 is announced as an add-on product for 3D design, surface modeling and NC programming. Mini computers with much more power at less cost started to appear. This was a major step forward and by 1984 the technology began to be competitive with traditional methods. For many years aircraft had of course been designed using computers, but now it was becoming possible to economically design saucepans and other domestic products with complex 3D shapes using a computer. Autodesk was founded by sixteen people in April 1982 in California by initiative of John Walker in idea to create a CAD program for a price of $1000 to can run on PC. John Walker has been running Marinchip Systems for two years before. In November at COMDEX trade show in Las Vegas was demonstrated the first CAD program in the world that runs on PC. This was the initial release of AutoCAD and deliveries begun in December. • 1983: Unigraphics II introduced to market • 1984, a Hungarian physicist, Gabor Bajor, smuggled two Macs into his country. At the time, ownership of personal computers was illegal under Communist rule. Using Pascal, he and a teenager, Tamas Hajas worked to write a 3D CAD program for the Mac which will be the beginning of Graphsoft Company. Drafting capabilities are added to CATIA in 1984, enabling it to function independently of CADAM. The first Autodesk Training Centre. In October AutoCAD version 2 (Release 5) with text improvements, DXFIN and DXFOUT commands, new Inquire commands, Object Snap, named views, Isometric capabilities and new Attribute features. • 1988: Surfware Inc., ships the first version of SurfCAM, a CAD/CAM program. • 1989: Parametric Technology ships the first version of Pro/ENGINEER.

  40. The History of CAD (1990-1995) • 1990:McDonnell Douglas (now Boeing) chooses Unigraphics as the corporate standard for mechanical CAD/CAM/CAE. Autodesk ships Animator Pro, a 2D painting and animation program for DOS. By 1993 over 15,000 copies have been sold worldwide. • 1991: Microsoft developed Open GL for use with Windows NT. Open GL is an API procedural software interface for producing 3D graphics and includes approximate 120 commands to draw various primitives such as points, lines, and polygons. Also includes support for shading, texture mapping, anti-aliasing, lighting and animation, atmospheric effects such as fogging and simulation of depth-of-field. Open GL, developed by Silicon Graphics, is a standard for the 3D color graphics programming and rendering. • 1992: Autodesk ships 3D Studio version 2 for DOS.Autodesk ships AutoCAD Release 12 for DOS in June. Includes AutoCAD SQL Extension (ASE)/Autodesk SQL Interface (ASI) that lets you establish links between AutoCAD and an SQL database. Advanced Modeling Extension (AME) release 2.1 is supported by Release 12, with region modeling and new solid primitives. AutoCAD Render is included with AutoCAD. • 1993: The first AutoCAD (Release 12) for Windows platforms. It required 8 MB RAM and 34 MB Hard Drive space for complete installation. The Windows version of AutoCAD includes 36 icons toolbox, allows multiple AutoCAD sessions, separate Render window, support for Windows GUI, DDE and OLE, as well as Drag-and-Drop and Bird's Eye view capabilities. The AutoCAD main menu has been eliminated; After initial configuration, AutoCAD displays the graphics screen. AutoCAD 12 for Windows was one of the most successful CAD programs ever • 1994: MiniCAD version 5. Hewlett Packard ships version 3.5 of PE/Solid Designer, its high end Solid Modeling. • 50,000 seats installed to date. • 1995: CATIA-CADAM AEC Plant Solutions are announced. This next generation object-oriented plant modeling system enables powerful knowledge-based engineering capabilities that can dramatically streamline the process of plant design, construction and operation. It brings the power of "smart" applications to the desktop with next generation object-oriented modeling. IDEAS Master Series version 2.1 from SDRC. Mazda Motors Corp. will install 2,400 seats of this product. Parametric Technology ships Pro/E version 15, the first parametric modeling CAD/CAM program and the first high-end 3D solid modeling package available on NT platforms.

  41. The History of CAD (1996-99) • 1996: Solid Edge version 3 from Intergraph hits the market at the price of around USD 6000. SolidWorks Co. ships Solid Works, an ambitious 3D package based on Parasolids modeling Kernel. It comes with a good complex surface modeling and a good graphical user interface. 3D/EYE Inc., ships Tri Spectives Technical version 2, a modeling, illustration and animation program for Windows platforms, at a very low price. Lightscape version 3, a high-end rendering and animation package, comes with IES photo-metric data capabilities. IES (Illuminating Engineers Society) is the industry standard for describing the shape and intensity of light energy distribution froma light source, ray tracing, natural light according to location and orientation of the building. Lightwave 3D version 5 and 5.5 from New Tek, a high-end rendering, modeling and animation program. AutoCAD LT 95. Diehl Graphsoft released MiniCAD 6 for Windows, the first cross-platform version of MiniCAD. Pro/E version 17 with a new module which allows files to be exported into VRML file format for display on the Internet. • 1997: Autodesk ships 3D Studio MAX release 2 and a cut-down version called 3D Studio Viz. EDS introduces a number of new industry-leading capabilities with its new version of Unigraphics, including WAVE - which will enable the definition, control and evaluation of product templates - considered the most important new technology affecting the CAD/CAM/CAE industry in the next five years. First version of IDEAS Artisan Series from SDRC, fully compatible with Master Series, priced at ~ USD 5,000. Form Z, a solid and surface modeler, first available only for Mac platforms, debuts on Windows market. • 1998: Autodesk Architectural Desktop - integrated architectural solution based on AutoCAD 14.First version of IronCAD for VDS market. Autodesk ships 3D Studio MAX version 2.5Lightwave 3D version 5.6 from New Tek, comes with Procedural shades for snow, water and rust, Stereoscopic rendering, SkiTracer image warping for real time visualization of generated sky, and more.Solid Edge version 3 from Intergraph with more than 150 new features.Solid Works 98 adds 150 new capabilities. • 1999: CATIA Version 5 for native Windows NT and UNIX. Lightwave 3D version 6 from New Tek.Think3 entry in the CAD market with thinkdesign, the first mechanical design software product to offer the power of parametric solids, advanced surfacing, wireframe and 2-D drafting, all in one environment.VectorWorks replaces MiniCAD.3D Studio MAX cumulus 29% of the entire 3D-animation market and 38% of the 3D PC markets.

  42. From Euclid to Desargues • Euclid's Optica, c. 300 B.C., was the first text on geometrical optics, in which are defined the terms visual ray and visual cone. • Vitruvius' Ten Books on Architecture which appeared about 25 B.C., was the only book on architecture to survive from antiquity. It profoundly influenced Renaissance architecture and thinking, including that of Alberti, who quoted Vitruvius in his Della pittura. Vitruvius wrote: Perspective is the method of sketching a front with the sides withdrawing into the background, the lines all meeting in the center of a circle. Unfortunately he didn't elaborate on that. Elsehere, Vitruvius' reference to Greek and Roman stage design, implied an understanding of the vanishing point. • Ptolemy's Optica, c. 140 A.D., was another early text on geometrical optics, and included theories on refraction. The centric ray is defined by Ptolemy as the ray that does not get refracted. The centric ray, we'll see, is important in the theory of perspective. In his Geographia, c. 140 A.D., Ptolemy applies the principles of geometric optics to the projection of the spherical surface of the earth onto a flat surface, to produce a map. He is said to have made the first known linear perspective construction for drawing a map of the world. Ptolemy apparently knew about perspective, but applied it only to maps and to stage designs. • Galen's De usu partium, c. 175 A.D., contains an early but erroneous description of how the eye creates images. The book was still important, however, as a stepping stone in the development of the theory of perspective. • From Islam, Alhazen's Perspectiva, c. 1000 A.D., was an important compendium on optics. It integrated the works of Euclid, Ptolemy, and Galen. • Roger Bacon's Opus Majus, c. 1260 A.D., included a section on optics, whose geometric laws, he maintained, reflected God's manner of spreading His grace throughout the universe. • John Pecham's Perspectiva communis, c. 1270 A.D., was another treatise on optics that was widely available during the Renaissance. • Blasius of Parma's Quaestiones perspectivae, c. 1390 A.D., was a popular adaptation of the works of Bacon and Pecham.

  43. From Euclid to Desargues • We are all familiar with Euclidean geometry and with the fact that it describes our three-dimensional world so well. In Euclidean geometry, the sides of objects have lengths, intersecting lines determine angles between them, and two lines are said to be parallel if they lie in the same plane and never meet. Moreover, these properties do not change when the Euclidean transformations (translation and rotation) are applied. Since Euclidean geometry describes our world so well, it is at first tempting to think that it is the only type of geometry. (Indeed, the word geometry means “measurement of the earth.”) However, when we consider the imaging process of a camera, it becomes clear that Euclidean geometry is insufficient: Lengths and angles are no longer preserved, and parallel lines may intersect. • Perspective is an example of the geometric operation of projection and section where projection lines from the outline of an object to the eye are sectioned or cut by a picture plane. This has roots in the conic sections, where projection lines from a circle to a point form a cone, which is then sectioned by a plane to give a circle, ellipse, parabola, or hyperbola, depending on the angle of the cutting plane. These ideas were expanded by Gerard Desargues (1593-1662), architect/engineer, into the branch of mathematics called projective geometry. • Projective geometry is a branch of mathematics that deals with the relationships between geometric figures and the images, or mappings, of them that result from projection. Common examples of projections are the shadows cast by opaque objects, motion pictures, and maps of the Earth's surface. Projection of one line onto another Central projection of one plane on another

  44. Frank Lloyd Wright • Wright used nature as the basis of his geometrical abstraction. His objective was to conventionalize the geometry which he found in Nature, and his method was to adopt the abstract simplification which he found so well expressed in the Japanese print. Therefore, it is not too shocking perhaps that in this quest his work should foreshadow the new mathematics of nature first put forth by Benoit Mandelbrot: fractal geometry. • --Leonard K. Eaton Floor plan from a late Wright residence.

  45. Platonic Solids • The so-called Platonic Solids are regular polyhedra. “Polyhedra” is a Greek word meaning “many faces.” There are five of these, and they are characterized by the fact that each face is a regular polygon, that is, a straight-sided figure with equal sides and equal angles. • The Greeks, who were inclined to see mathematics as something of a religious truth, found this business of there being exactly five Platonic solids very compelling. The philosopher Plato concluded that they must be the fundamental building blocks – the atoms – of nature, and assigned to them what he believed to be the essential elements of the universe. He followed the earlier philosopher Empedocles in assigning fire to the tetrahedron, earth to the cube, air to the octahedron, and water to the icosahedron. To the dodecahedron Plato assigned the element cosmos, reasoning that, since it was so different from the others in virtue of its pentagonal faces, it must be what the stars and planets are made of.

  46. references • General Science and Art: http://library.thinkquest.org/3257/ • Digital Design Media: http://www.gsd.harvard.edu/~malcolm/DDM/GALLERY/15.01_1956.gif • Durer: http://www.newcastle.edu.au/department/fad/fi/woodrow/durer-c.htm • General Information on Perspective: http://www.newcastle.edu.au/department/fad/fi/woodrow/an-persp.htm • Leonardo: http://www.mos.org/sln/Leonardo/LeonardosPerspective.html • Alberti: http://www.leonet.it/culture/nexus/98/Pasquale.html • Pantheon: http://www.leonet.it/culture/nexus/98/Sperling.html • Palladio (Stephen Wassell): http://www.leonet.it/culture/nexus/98/Wassell.html • Brunelleschi: http://www.cuny.edu/multimedia/arsnew/arch1.html

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