Math 0025 Geometry and Art course page, Fall 2025


Prof: Robin Pemantle, DRLB 4N22, phone: 215-898-7894, pemantle@math.upenn.edu , Zoom 932 808 9909
TA: Hanan Jannoud, hjannoud@sas.upenn.edu , Office hours Mondays 5-6pm in DRLB 4C9

Course Description

Book: "Perspective and Projective Geometry" by Crannell, Frantz and Futamura (Princeton University Press, 2019).


In stock at the University Bookstore, or order it at this link:
link to Princeton University Press website for the book (use code BLOOM50 at checkout) - wait, this code appears to be broken, use the Penn Bookstore for now!
Please buy the book, don't rent it. Some of the pages are meant to be drawn on or written on.


Syllabus

Calendar at a glance         Final exam: Friday December 12, 12pm-2pm in DRL 4C6

Math resource

Major topics superseded by: Review

Groups for time period starting MON Nov 3




PDF file of article by Zeke Emmanuel, Penn Today, 25 August 2025        html version (if you can get past the NYT paywall)

First short assignment, due Wednesday September 3, on paper at the beginning of class.

Some window taping photos

Left window
Right window
Left window ; tape only
Right window ; tape only

Definitions we discussed in class SEP 3
Special case of vanishing points in the case where a line is parallel to the picture plane
Solution to Problem 1.8 in Chapter 1, written somewhat formally

Homework problems from Chapter 1 are due Wednesday September 17 at the beginning of class.

Blank copy of the letter T in case you need to print another one to work on

Solutions to the T problem

Homework problems from Chapter 2 are due Wednesday September 24 at the beginning of class.

Homework problems from Chapter 3 are due Wednesday October 01 at the beginning of class.

Ceva's theorem: good and bad proof writeups, with corrections shown COMING AFTER WE FINISH THIS IN CLASS, probably MON 9/29 afternoon

Homework problems from Chapter 4 are due Wednesday October 08 at the beginning of class. DO NOT DO problem P4.2.3. Instead, do the problem found at this link: replacement problem

Homework problems from Chapter 5 are due Monday October 20 at the beginning of class. A few students may need to do a redo for half credit on Problem 4.3 replacement; this will be due along with the Chapter 5 homework.

An optical illusion created using perspective techniques:
Borromini's colonnade in the Palazzo Spada
click here for the Wikipedia entry

First installment of homework problems from Chapter 6 are due Wednesday October 29 at the beginning of class:
Problems E6.1 - 6.3 and P6.1 - 6.4.

Second installment of homework problems from Chapter 6 are due Wednesday November 5 at the beginning of class:
Problems E6.4 - 6.6, A6.1 (photography) and the extra problem on trisection, posted at this link

Selection of problems from Chapter 7 due Wednesday November 12 at the beginning of class: Problems E7.1.1, E7.1.2 and A7.1.1.
For problem E7.1.2: Carefully complete the drawing along the lines we designed in class. Write a precise account of the construction at each step.
Please leave all construction lines visible, drawn lightly, then darken the line segments forming the edges of the kite shadow and shade in the kite.

Homework problems from Chapter 9 are due Monday November 24 at the beginning of class. For E9.2 you may assume that the picture is in one-point perspective; if you do, please indicate in your solution where you use this assumption. Please also indicate in your solution where you use the assumption that the tabletop is a square.

Final project instructions

This selection of homework problems from Chapter 10 is now due Monday December 8 at the beginning of class. For hints, see below.
    Problem E10.1 (inscribed square)
    Problem E10.2 (octahedron)
    Problem A10.1 but you only need to do 2 letters, not 4.

Click here for the hints on E10.1 and E10.2. You will need to turn in a sketch and written description that does the problem, or as much of it as you can, as you normally would. That is, weave together sketching and justifying into a narrative answer, rather than just listing answers to the hints.

"A car, a bike and a truck" review problem

Review questions using student art

Practice final exam questions   Problem 1 solution   Problem 2 solution   Problem 3a solution   Problem 3b solution   Problem 4 solution   Problem 5 solution