|First Year Seminar: for instructors of Mathematics of Art|
- The Summer assignment
- The Syllabus
- A list of supplies to buy
- The weekly schedule
- The list of one-page papers
writing and grading art are two things that math instructors
don't often do. Here are some of the tools I use that help
me do each.
2. Some things that might be helpful to instructors of math and art
|Projective Geometry applied to Perspective art: A proofs-based course|
Taping: the After Math
||• basic understanding of parallel lines and
• images of lines in a picture plane;
• plan views;
• definition of vanishing point;
• importance of the notion of “parallel” in determining the existence and location of the vanishing point.
||• problem solving
• artistic application of the rules “Lines parallel to the picture plane have parallel images; lines not parallel to the picture plane but parallel to one another converge to the same vanishing point”.
of a line
||Students explore possible definitions of
projection of points and lines onto a plane. Topics:
• visualizing projections of points and lines in R2 and R3;
•considering “special cases” of projection;
•understanding the difference between artistic applications and mathematical definitions.
lines and points
||• introduction to GeoGebra
• review definition of vanishing points
• motivate notion of “points at infinity”
||• The axioms of the extended plane and extended
• logical consequences of those axioms;
• proving statements by using axioms and definitions;
• visualizing points and lines at infinity.
and Mesh Maps
||Students explore the basic structure of a mesh
and a formal definition of projection onto a plane. As an
application, they draw and devise solutions to geometric
The notion of a mesh is not a standard one in projective geometry. A mesh is useful, however, in artistic applications such describing a 3-d object like a house: for this object, we want to draw some lines (such as the top edge of the roof) but not others (we don’t draw the line that connects the front top of the roof to the back lower left corner of the house). Similarly, for this object we care about some points of intersection (the four corners of the floor) but not others (the front left edge of the roof does not intersect the back right edge of the roof in a point in R3, even though their images intersect in Figure 2).
||Fence division problems and their relation to
Division (follow up)
||A power-point developed for our minicourse
participants to expand on the worksheet above
|8. Squares in
||Students discover the viewing circle; they
locate the viewing target and viewing distance for a
square in two-point perspectives. From there, they learn
to draw a box and then a cube in 2-point perspective.
We foreshadow the need for harmonic sets of points.
and Streetlight page
|• How to draw shadows;
• defining perspective from a point and perspective from a line;
• exploring Desargues’s Theorem.
|9. Desargues Proof
||In which we prove the theorem. (Uses a
"modified Moore-method" approach).
||An intro to colineations, in particular
perspective colineations in the plane
||We show that period-2 homologies (like
reflections) relate to the above worksheet on squares in
||We practice constructing elations; these are
like the fencepost-repeating problems above
Cross ratio and the vanishing point