Post on 21-Jan-2015
description
transcript
Albrecht Dürer: Art, Geometry, and Printing
"Without [the art of measurement] no one
can become a true artisan."
Albrecht DürerThe Painter's Manual
(1525)
Welcome to the Renaissance!
Vasari's Lives of the Artists: "Rebirth" of classical forms.
But: What are classical forms?
Medieval or Classical?
Notre Dame, Paris Crude, barbaric.
("Gothic", but actually, medieval)
Pantheon, Rome Domes, columns
Medieval or Classical?
Book of Hours (Duke of Berry)
Non-realistic
Boscoreale (Naples) Realistic depiction of
space
Medieval or Classical?
Merovingian minuscule.
"Form of torture"
Latin capitals. Easy to read.
A Quest for Lost Arts
How did the Romans and Greeks make their art?
Many artifacts, but few documents.
Reverse engineering!
Raphael's School of Athens (1511)
A Clasical Education
"Libera" (free men) vs. "Manus" (hand work) Plato's educational system (The Republic)
Arithmetic (number properties) Geometry (Euclidean) Music (as applied arithmetic) Astronomy (as applied geometry)
Math is all the liberal arts
The Degeneration of Philosophy
Martianus Capella (fl. 420) The Marriage of Philo and Mercury Geometry reduced to areas, lengths, geography
Boethius (fl. 500) Quadrivium Basis for medieval education
The Rebirth of Western Mathematics
Islamic Civilization Preserved and extended Greek geometry Abu'l-Wafa On the Parts of Geometry Necessary for
Artisans Geometric designs
The Spanish Connection Campanus of Novara (fl. 1300)
New edition of Elements Geometry more than lengths and areas
The Mathematization of Art
Cathedral Building Jordanus Nemorarius
(fl. 1225): On Weights Filippo Brunelleschi
(1377-1446) Architect of duomo Perspective Art is applied math!
Leon Battista Alberti (1404-1472)
On Painting (1435) Painting as a liberal art Mathematical basis
On Architecture (1452) Study of Roman ruins,
Vitruvius Importance of ratios and
proportionSanta Maria Novella (Florence)
Vitruvius Ten Books on Architecture
Inspired Alberti, Pacioli, da Vinci
Importance of ratios Face: Body = 1:10 Arms: Legs = 1:1 Foot: Body = 1:6 Forearm: Body = 1:4
Aesthetics quantifiable!
da Vinci's "Vitruvian Man"
Every Artist a Geometer...
Piero della Francesca (1415-1492) "Flagellation" one of
oldest to use true perspective.
Treatise on the Abacus Five Books on the
Regular Solids.
...and every Geometer an Artist
Luca Pacioli (1446-1517) Summary of
Arithmetic, Geometry, Proportion, and Proportionality (1494)
The Divine Proportion (1509), includes Francesca's Five Books (illustrated by da Vinci)
Nürnberg in the Fifteenth Century
Political center (Diet meetingplace)
Oldest paper mills in Germany
Printing center (Regiomonantus's press)
Weltchronik of Wolgemut
Albrecht Dürer
Born Nürnberg, 1471 Education of an Artist
Apprenticed to Wolgemut Carved wood blocks for print illustrations Two Italian trips (1494-5, 1505-7) Met Pacioli, studied Alberti, Euclid, Vitruvius
The Painter's Manual (1525)
"Nothing is more annoying to men of understand in than a blunder in a painting..."
Practical geometric construction for artists.
Theoretical geometry
Artist's Problem #1
How do you draw an object?
Composition of basic shapes.
Straight lines and circles...
...but other curves (e.g., helix) required for "realism."
Pope Benedict in Munich
Basic Drawing
Straight lines and circles easy to construct.
Paraphernalia of daily life often includes other shapes.
Crozier: Spiral and circular arcs.
Croziers from Metropolitan Mustum of Art
Dürer's Spirals
Divide AB into fourths at C, D, E.
Bisect D, F. Draw semicircles AB,
BC, CE, ED with centers alternating D, F (final semicircle DF with center at midpoint).
Non-Archimedean spiral.
Plan and Elevation
Representing space curves.
Dual viewpoint. Detailed instructions
on how to construct. Adapted from
architectural plans.
Conic Sections for Artists
The ellipse as a conic section.
Dürer invents German name ("egg-shaped line")
Similar constructions for parabola ("burning line") and hyperbola ("fork shaped line").
First contour maps?
Dürer's Polygons
Simple construction for regular hexagon: no need to change compass opening.
Join alternate vertices for equilateral triangle.
The Heptagon
Side of heptagon is half side of equilateral triangle.
Method given by Abu'l-Wafa (On the Parts of Geometry Necessary for Artisans)
Dürer's method new to Europe?
Kepler: "Obviously" too short.
How Close to Exact?
H is vertex of regular heptagon.
For circle with diameter 20 feet, approximation gives side of heptagon as 8 feet, 7.92 inches vs. actual side of 8 feet, 8.13 inches.
Pentagons
Bisect radius OA at C. Join CB, and mark CD equal to CB. BD is side of pentagon.
Method given by Ptolemy in Almgest
Exact construction.
Rusty Compass Pentagon
AB as side, draw circleswith centers A, B and radius AB.
Join CD. Draw circle with center D and radius AB.
Draw EG to H and FG to I. Make IJ, HJ equal to AB.
ABHJI is equilateralpentagon.
Also Approximate
Pentagon is equilateral but not equiangular
Method from Geometria Deutsch of Matthias Roritzer
"Rusty compass" problems appear in Islamic geometry (Abu'l-Wafa gave exact construction for pentagon).
Tiling Patterns
Floor covering using polygons
Regular tiling using squares, triangles, octagons.
Heptagonal Tilings Pentagonal Tilings
A Near Miss...
Pentagons Joined rosettes Fill space with other figures
Kepler (1619) Critiqued Dürer's heptagon No mention of Dürer tilings
Penrose (1974)
...Or Not
Problem: How do you represent three-dimensional objects schematically?
"Framework" to show structure (Leonardo's drawing of rhombi-cuboctahedron from Pacioli's Divine Proportion)
Dürer's Polyhedral Nets
Dodecahedron Rhombicuboctahedron
Mapping the World
Can "unwrap" sphere in same way (Dürer).
Spherical projection problem.
Waldseemüller map (1506) first to use designation "America."
Artist's Problem #2
How do you make a painting look like a real scene?
More distant objects look smaller (foreshortening).
Convergence of parallel lines (perspective).
A Building in Nürnberg
Foreshortening
Used to convey sense of distance.
Dürer used (St. Jerome woodcut)
Mathematical basis: Objects of equal size subtend smaller angles at greater distances.
Practical Design
Inscriptions on buildings and columns.
Equal arcs make for equal apparent size (from vantage point).
Modern street writings.
Perspective
Radial symmetry about vanishing point.
Not in Classical Art! Renaissance
invention.
Woodcuts and Paintings
Dürer learns "secret" during 1506 visit to Italy.
Explains technique in Painter's Manual.
Ratios of Gold...
Pacioli's Divine Proportion Ancients constructed
temples and art using key ratios.
Ratio between diagonal of pentagon and side is "Golden Ratio"
...and Dross
More natural to use Fibonacci sequence
ALL Vitruvian ratios rational.
NO evidence for deliberate use of side/diagonal of pentagon.
Pacioli doesn't use golden ratio as basis.
A Brief History of Typography
Roman majuscule (CAPITAL LETTERS). Merovingian minuscule (lower case letters,
joined with ligatures). t, e, i, r: er et te ti tr
Merovingian script, from Luxeuil manuscript.(From http://medievalwriting.50megs.com/scripts/examples/luxeuil.htm)
Carolingian Minuscule
Alcuin and Charlemagne Reform of Merovingian
script Basis for modern Roman
lower case Far easier to read and
write! Degeneration of Roman
type?
Manuscripts and Printers
Printing press China/Korea Print money, playing cards.
Gutenberg What should letters look
like? Challenge: Make an "A" using
the Golden Ratio (or Fibonacci Rectangle)
Classic Fonts
Felice Feliciano Pacioli
Letters in appendix to Divine Proportion
Based on circle and square
Thick limbs = 1/9 height, thin limbs = 1/2 of thick
NO golden ratio.
Dürer's Letters
Circle and square Ratios
Square into four parts Thick limbs = 1/10
height, thin limbs 1/3 of thick.
Top center of width. Base serifs use circle
with radius 1/7 of side of square
Gothic Letters, Old Style
Squares atop squares 1/3 division of sides Vertical diagonals Letters as
components of other letters ("i" as basis for u, b, etc.)
Gothic Letters, New Style
Old style, with additional serifs, squares.
Roman type only upper case letters.
Dürer first (?) to geometrize lower case letters.
Reactions
Aesthetics Cresci (1560): Curves of actual letters not circles
and straight lines. Letters "ugly"
Practicality No one learns to write this way! Difficult to cut type metal this
precisely Impression printing has no need
for mathematical defined fonts.
The Computer Connection
Bit map fonts Graininess problem One image for each size of each letter
True-Type fonts Letters defined by lines and curves Curves beyond circle (splines)
Printers (and computer screens) Ink-jet: True-Type to bitmap to print Laser printer: True-Type to laser pointer to print
Some Final Thoughts
Thinking componentwise and choosing what is important are essential skills in the creation of art...and mathematics.
The constructive arts are an underutilized resource in the teaching of mathematics.