358 12. MODERN GEOMETRIESFIGURE 4. Central projections preserve tangency, midpoints, and
(usually) parallelism.treatise on arithmetic and geometry in 1494, and a second book, De divina pro-
portioned in 1509. He gave the name divine proportion to what is now called the
Golden Section, the division of a line into mean and extreme ratios. Interest in the
five regular solids branched out into an interest in semiregular solids. Leonardo da
Vinci designed wooden models of these, which were depicted in Pacioli's treatise.
The regular and semiregular solids formed an important part of Diirer's manual
for painters, published in 1525. He showed how to cut out a paper model of a
truncated icosahedron, which consists of 12 pentagons and 20 hexagons (Fig. 5).
The solid, although not the name, has become very familiar to modern people
through its application in athletics and organic chemistry.
A geometric description of perspective was given by Leon Battista Alberti
(1404 1472) in a treatise entitled Delia picture., published posthumously in 1511. If
the eye is at fixed height above a horizontal plane, parallel horizontal lines in that
plane receding from the imagined point where the eye is located can be drawn as
rays emanating from a point (the vanishing point) at the same height above the
plane, giving the illusion that the vanishing point is infinitely distant. The appli-
cation to art is obvious: Since the canvas can be thought of as a window through
which the scene is viewed, if you want to draw parallel horizontal lines as they
would appear through a window, you must draw them as if they all converged on
the vanishing point. Thus, a family of lines having a common property (passing
through the vanishing point) projects to a family having a different common prop-
erty (being parallel to one another). Obviously, lines remain lines under such a
projection. However, perpendicular lines will not remain perpendicular, nor will
circles remain circles.