Visualizing 3D
Between Measurement and Illusion
Dan Collins
VizProto
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]
Euclid
300 B.C.
Pythagorean Theorem
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.
Euclid’s description of the Pythagorean theorem
Euclid
300 B.C.
Euclid also wrote Optica, the first
text on geometrical optics, in which
he defines the terms visual ray and
visual cone.
Raphael, The School of Athens, 1509, Fresco, Vatican, Rome
Detail showing Euclid with his students
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.
Pythagorus
580-520 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.”
Pantheon
Rome, Italy, 118 to 126 AD
Architect unknown
Exterior view of the Pantheon
in modern day Rome
Interior view of the Pantheon
Giovanni Paolo Panini, c. 1750
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" consonantsymphonic ratios. To generate
harmony, the laws of
arithmetic, geometry,
astronomy and musicalproportions are fused. It
"resembles the heavens", but
is a resemblance based on
mathematical knowledge, a
summary of the ancient
quadrivium*.”
--Girt Sperling
Section showing pythagorean ratios at work in the Pantheon.
* The quadrivium was the higher division of the
seven liberal arts in the Middle Ages, composed
of geometry, astronomy, arithmetic, and music.
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.
Brunelleschi
1377-1446
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.
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.
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
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.
Masaccio
1401-1428
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.
Alberti
1404 - 1472
Alberti’s “fenestre” (window) or “velo”
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
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.
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.
Piero della Francesca
c.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.
Leonardo
1452 - 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.
Dürer
1471-1528
A woodcut from Albrecht Dürer's treatise on measurement
Underweysung der Messung, 1527
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.
Dürer
1471-1528
Dürer's Perspectograph, early 16th c. (replica)
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 scenographers 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.
Palladio
1508-1580
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.
vitruvius
c. 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.
Hans Holbein, The Ambassadors, 1536
Living Emblem of the United States
Marines. 100 officers and 9000
enlisted men. Marine Barracks,
Parris Island, S.C.; Brigadier
General J.H. Pendleton,
Commanding. Mole & Thomas,
Chicago Illinios, 1919.
Francesco Borromini
1599 - 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.
Diderot
1713 - 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)
Francois Willeme - Photosculpture
1860
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 regularlyspaced 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
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.
Etienne-Jules Marey
1830- 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 penrecording device to plot the traces over time.
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.
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.
Dynamic Visualization
Visualization of Storm
patterns combines 3D
graphics and actual
metrics
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).
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.
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.
http://www.youtube.com/watch?v=bBQQEcfkHoE
http://www.youtube.com/watch?v=khn1lPdy68M
http://www.youtube.com/watch?v=bBQQEcfkHoE
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.
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.
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 standalone 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.
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.
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 2.5 Lightwave 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.
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.
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
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.
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.
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
Visualizing the Web
Gunilla 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.”