Perspective and Projective Geometry

Published in 2019 by Princeton University Press, an IBL textbook that helps you to discover connections between projective geometry and perspective drawing.

Sketch of concept

It began with some sketches of an idea - having taken students to the Blanton Museum of Art to find mathematics in artwork, I was inspired to create a two point perspective drawing of a gallery space with works of art depicting some ideas from the book. I first tried photographing shadows in actual museums, but the lighting was never quite right.

GeoGebra to hand drawing

I first created the mathematically correct drawing in GeoGebra, so that I could more easily fiddle with lines and vanishing points to get it just right. I wanted to make sure the vanishing points were not too far off the page, while at the same time being mindful of the composition.

Almost completed drawing

This is the drawing without the addition of Taylor’s drawing on the wall. I left many of the construction lines to show how it was made.

Book cover

I got to draw the cover of the book! The drawing, done in black and white charcoal on gray paper, is of an imaginary art gallery with artwork that demonstrates double Desargues’ theorem (Roger Jorgensen-esque statue with shadow) and cross ratios (Donald Judd-inspired cubes). On the wall is a copy of mathematician and artist Brook Taylor’s drawing from his book on perspective.

current positions

Professor of Mathematics at Southwestern University
holder of the John H. Duncan Chair endowed by the Brown Foundation
Board member of the Japan-America Society of Greater Austin (JASGA)

Previous positions

Associate and Assistant Professor of Mathematics at Southwestern University, 2007-2018
Lord Chair of Mathematics and Computer Science 2019-2022
Chair of the Department of Mathematics and Computer Science 2018-2020
Chair of the Diversity, Inclusion, Belonging and Equity (DIBE) Task Force 2022-23
Chair of the Curriculum Committee 2017-2018
Chair of the Enrollment and Retention Committee, 2015-2016
Interim co-President of Japan-America Society of Greater Austin (JASGA), 2022
Co-founder, co-organizer and co-instructor of SU EQUIP Summer Program 2016-2018
Science Advisor, NPR's Studio 360, Science and Creativity series, 2015-2016
Advisory board member, Art.Science.Gallery in Austin, TX, 2013-2016

Teaching

Linear Algebra and Modern Calculus I in Fall 2024
for more information, click here.

community outreach and selected talks

View the full lesson here.

2023 Mathematical Problem Solving and Discovery Through Perspective Drawing! Santa Cruz Math Teacher’s Circle, Santa Cruz, CA
2022 How to mathematically immerse yourself in a perspective drawing, UT Austin Math Teacher’s Circle, Austin, TX
2022 Curious invariants in projective geometry, and where to find them in art and music, Rice Undergraduate Colloquium, Rice University, Houston, TX
2019 Perspectives of a Japanese female mathematician artist, inaugural talk, WISE (Women in Science and Engineering) Club at Anderson High School, Austin, TX
2018 Visualizing homographies: distortions (or the lack thereof) in photos shadows and art, Paideia Connections Talk, Southwestern University
2018: Perspectives of a mathematician artist, Honors Summer Math Camp Colloquium for high school students, Texas State University
2018: How to mathematically immerse yourself in art, University Lecture Series, Texas State University
2018: When artists become mathematicians, public lecture, Phi Beta Kappa (En)Lightning Talks Houston
2018: Learning station on hyperbolic crochet coral reefs, Hot Science, Cool Talks, Environmental Science Institute, UT Austin
2017: Fractals in Japanese woodblock prints, Academic Lecture Series, Japan America Society of Greater Austin
2017: The Mathematics of sidewalk illusions, TED-Ed Animation and Lesson
2016: Learning station on hyperbolic crochet: Fuzzy Math, Thinkery21 at the Thinkery, Austin, TX
2016: Learning session: Mathematics of perspective drawing, Operation Math Girls Conference for high school students, Sam Houston State University, TX
2016: Illustrating Abstract Mathematics Through Perspective Drawing, SU Art History Program's Representations Lecture Series, Southwestern University
2016: Finding the Viewpoint at a Museum: a How-To Guide, Joint Mathematics Meetings, Seattle, WA
2015: How to Mathematically Immerse Yourself in a Work of Art, Infinite Horizons Lecture Series, Kennesaw State University, GA and Mathematics and Statistics Colloquium, Sam Houston State University, TX
Dec 2014: Community workshop: Hyperbolic crochet, Art.Science.Gallery, Austin, TX
Dec 2014, 2013: Community workshop: Mathematical origami ornaments, Art.Science.Gallery, Austin, TX

PUBLICATIONS

S. Friday, F. Futamura, J. Smith, A. Waclawczyk, Powers of defective matrices from diagonalizable dilations, Linear Algebra Appl., 661 (2023) 202-221.
TEXTBOOK: A. Crannell, M. Frantz, F. Futamura, Perspective and Projective Geometry, Princeton University Press, 2019.
A. Crannell, M . Frantz, F. Futamura, An (isometric) perspective on homographies, J Geometry and Graphics, 23 no. 1 (2019) 65-83.
A. Crannell, M. Frantz, F. Futamura, Factoring a homography to analyze projective distortion, J Math Imaging Vis, 61 no. 7 (2019) 967-989.
F. Futamura, A. Marr, Taking Mathematics Abroad: A How-To Guide, PRIMUS, 28 no. 9 (2018) pp. 875-889.
F. Futamura, R. Lehr, A new perspective on finding the viewpoint, Mathematics Magazine, 90 no. 4 (2017) pp. 267-277.
- Student research with Robert Lehr, class of 2015
- Recipient for the 2018 MAA Carl B. Allendoerfer Award for expository excellence of papers published in Mathematics Magazine in 2017.
A. Crannell, M. Frantz, F. Futamura, The Image of a Square, Amer. Math. Monthly, 124 no. 2, (2017) pp. 99-115.
A. Crannell, M. Frantz, F. Futamura, The Cross ratio as a parameter for Dürer's solid (Party game for a 500th anniversary) J. Math. Arts, 8 no. 3-4, (2014), 111--119.
- On the shortlist of four papers considered for the 2016 JMA Outstanding Paper Award
A. Crannell, M. Frantz, F. Futamura, Dürer: Disguise, Distance, Disagreements, and Diagonals! Math Horizons, 22 no. 2, (2014) 10-11.
- Reprinted in: The Best Writing on Mathematics 2015, Princeton University Press, 2015.
R. Denman, F. Futamura, K.C. Richards, On sharp frame diagonalization, Linear Algebra Appl., 438 no. 5, (2013) 2210-2224.
F. Futamura, Frame diagonalization of matrices, Linear Algebra Appl., 436 no. 9, (2012) 3201-3214.
F. Futamura, Localizable operators and the construction of localized frames, Proc. Amer. Math. Soc., 137 (2009) 4187-4197.
F. Futamura, Symmetrically localized frames and the removal of subsets of positive density, J. Math. Anal. Appl., 326 no. 2, (2007) 122-1235.
F. Futamura, Banach framed, decay in the context of localization, Sampl. Theory Signal Image Process., 6 no. 2, (2007) 151-166.

AWARDS AND GRANTS

2018: MAA Carl B. Allendoerfer Award for expository excellence of papers published in Mathematics Magazine in 2017
2015: Dr. Lawrence Musgrove’s Visual Thinking for Engaged Learning grant, $500, funded by the Southwestern University Presidential Innovation Grants Initiative
2012-2014: NSF-DUE Type I grant #1140113, $69,432, a two-year grant to write a textbook on Projective Geometry Applied to Perspective Art (together with co-PI Annalisa Crannell's portion, $73,344), National Science Foundation
- One of 125 awards out of 1050 submitted proposals.
2011: AIBL Small Grant, $4000, a grant to create an inquiry based learning math and art course, funded by the Educational Advancement Foundation
2009: Faculty-Student Project Research Grant, $7590, a grant to do research with a student for his honors thesis for 8 weeks, Southwestern Univerisity
2009: Brown Innovation in Teaching Award, $9000, a grant to research innovative pedagogy and organize a teaching workshop, Southwestern University
2007: B.F. Bryant Prize for Excellence in Teaching, Vanderbilt University