First Year Seminar: for instructors of Mathematics of Art

1. Things my students see  

- The Summer assignment
- The Syllabus
- A list of supplies to buy
- The weekly schedule
- The list of one-page papers

2. Some things that might be helpful to instructors of math and art

Grading 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.

    Grading the writing projects.
    Grading the art projects.

Projective Geometry applied to Perspective art: A proofs-based course

The materials below are some of the materials being developed as
a collaborative project by Marc Frantz in Indiana, Fumiko Futamura in Texas, and Annalisa Crannell in Pennsylvania.   This project is supported by NSF TUES Grand DUE-1140135, so thank you for your tax dollars!

Projective Geometry applied to Perspective Art is an inquiry-based course designed for sophomore- and junior-level mathematics majors.  These materials in particular are ones that Crannell has used in her own course.  Because they are still in draft form, comments are welcome! 
Please feel free to contact us!

We presented an MAA Minicourse in January 2014; here are the slides and handouts from that minicourse.

In-class Worksheets 
1.  Window Taping: the After Math
• basic understanding of parallel lines and planes;
• 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.
2. Drawing ART
• 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”.
3. Image 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.
4. Geogebra lines and points
• introduction to GeoGebra
• review definition of vanishing points
• motivate notion of “points at infinity”
5. Extended Real Space
• The axioms of the extended plane and extended real space;
• logical consequences of those axioms;
• proving statements by using axioms and definitions;
• visualizing points and lines at infinity.
6. Meshes 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 division problems.

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).
7.  Geometric Division
Fence division problems and their relation to the cross-ratio
7. Geometric Division (follow up)
A power-point developed for our minicourse participants to expand on the worksheet above
8. Squares in 2-point perspective
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.
9.  Desargues Discovery
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).
10.  Colineations
An intro to colineations, in particular perspective colineations in the plane
10. Harmonic Homologies
We show that period-2 homologies (like reflections) relate to the above worksheet on squares in 2-point perspective.
10.  Elations
We practice constructing elations; these are like the fencepost-repeating problems above
11. Invariants:  Casey Angles

11.  Invariants:  Circular products

11:  Invariants: Cross ratio

11:  Invariants:
Cross ratio and the vanishing point

This page is maintained by Annalisa Crannell, Franklin & Marshall College.  Last updated July 2014.