RICHARD HAMMACK
Virginia Commonwealth University
Department of Mathematics and Applied Mathematics
1015 Floyd Ave.
Richmond, Virginia 23284-2014
Office: 4166 Harris Hall
Email: rhammack @ vcu . edu
Webpage: https://richardhammack.github.io/
YouTube: @StraightedgeAndCompass
EDUCATION:

PhD in Mathematics, 1996

University of North Carolina
at Chapel Hill

MS in Computer Science, 1988

Virginia Commonwealth University
Richmond, Virginia


BFA in Painting, 1983

Rhode Island School of Design
Providence, Rhode Island

EMPLOYMENT:
  • Professor, Virginia Commonwealth University, 2015-present
  • Faculty: Epsilon Camp, Summer 2024
  • Associate Professor, Virginia Commonwealth University, 2006-2015
  • Associate Professor, Randolph-Macon College, 2005-2006
  • Assistant Professor, Randolph-Macon College, 2000-2005
  • Visiting Assistant Professor, Wake Forest University, 1997-2000
  • Instructor and GTA, University of North Carolina at Chapel Hill, 1989-1997
  • Instructor, Virginia Commonwealth University Dept. of Mathematical Sciences, 1988-1989
  • GTA, Virginia Commonwealth University Dept. of Mathematical Sciences, 1986-1988
  • Field work and cartography, VCU Archaeology Research Center, 1984-1986
Employment Prehistory: Excavating a 19th century bateau in Richmond's Kanawha Canal basin with VCU's Archaeology Research Center, summer 1985.
BOOKS:

 

Handbook of Product Graphs, Second edition
by Richard Hammack, Wilfried Imrich and Sandi Klavzar

CRC Press - Taylor and Francis Group
Series: Discrete Mathematics and its Applications
2011.

Click on title for more information.

Book of Proof (3rd edition)
by Richard Hammack
2018.

Self-published open textbook

Click on title for more information.
CHAPTERS:

 

Chapter 10: Digraph Products
Classes of Directed Graphs

Editors: J. Bang-Jensen & G. Gutin
Springer
Series: Springer Monographs in Mathematics
2018.

 Section 8.10 Graph Families
RH, W. Imrich and S.Klavzar, et al.
Handbook of Discrete and Combinatorial Mathematics
Second Edition

Editors: K. Rosen, D. Shier, W. Goddard
Chapman and Hall/CRC
Series: Discrete Mathematics and its Applications
2017.
AWARDS:
  • Recipient: Reuben Farley Award for Innovative Teaching and Excellence, VCU,  2024
  • Recipient: Best Photograph, painting or print, JMM Mathematical Art Exhibit, Boston, MA 2023
  • Recipient: VCU College of Humanities and Sciences Distinguished Teaching Award, 2020
  • Recipient: UNC Tanner Award For Excellence in Undergraduate Teaching, 1997
  • Recipient: UNC J. Burton Linker Award for Excellence in Teaching, May 1993
  • Nominee: VCU College of Humanities and Sciences Distinguished Teaching Award, 2018
  • Nominee: VCU College of Humanities and Sciences Distinguished Teaching Award, 2017
  • Nominee: VCU College of Humanities and Sciences Distinguished Lecturer Award, 2015
  • Nominee: VCU College of Humanities and Sciences Excellence in Scholarship Award, 2013
  • Nominee: Tanner Award For Excellence in Undergraduate Teaching, 1994
  • Nominee: Tanner Award For Excellence in Undergraduate Teaching, 1993
JURIED MATHEMATICAL ART EXHIBITIONS:
  1. 4 Views of the 4-Cube, in MathArt at MathFest, Sacramento, CA, August 6-9, 2025.
  2. Video: Why The Tracks Converge, screened in Bridges Short Film Festival, Eindhoven, Netherlands, July 17, 2025.
  3. 4 Views of the 4-Cube, in Intersections: Math Art Truth Humanity, Mercer Island Community and Events Center, Mercer Island, WA, March 3-April 25, 2025.
  4. The Hypercube Pop-Up Book, in Joint Mathematics Meetings Mathematical Art Exhibit, Seattle, WA, January 8-10 2025.
  5. 5-Dimensional Box, in Bridges 2024 Mathematical Art Exhibit, Virginia Commonwealth University, Richmond, VA, August 1-5 2024.
  6. Topological Envelopes, in Seeing the Unseen: Math and Art, Wignall Museum of Contemporary Art, Chaffey College, January 8-March 9, 2024.
  7. 4 Views of the 4-Cube, in Joint Mathematics Meetings Mathematical Art Exhibit, San Francisco, CA, January 3-6, 2024.
  8. Mathematical Pop-ups (with Vic Bednar) in Bridges Mathematical Art Exhibit, Halifax, Nova Scotia, July 26-August 1, 2023.
  9. Topological Envelopes, in Joint Mathematics Meetings Mathematical Art Exhibit, Boston, MA, January 4-7, 2023.
  10. Closed Surface Envelopes, in Bridges Mathematical Art Exhibit, Aalto University, Helsinki Finland, August 1-5, 2022.
  11. Unit Circle Surface, in Hypergraphics Invitational, Rhode Island School of Design and Brown University, October 1984.
RESEARCH PUBLICATIONS:
  1. Hypercube models via genus embeddings, Bridges 2024 Proceedings, 2024.
  2. Euler's theorem for regular CW-complexes (with Paul Kainen), Combinatorica,  44(3) 453-465, 2024.
  3. Automorphism groups of half cubes (with Benjamin MacKinnon), Involve, 16(2) 321-329, 2023.
  4. A new view of hypercube genus (with Paul Kainen), American Mathematical Monthly, 128:352-359, April 2021.
  5. Factorization of Platonic polytopes into canonical spheres (with Paul Kainen), Geombinatorics Quarterly, 31(1) 5-9, 2021.
  6. Palindromic products (with Jamie Shive),  Art of Discrete and Applied Mathematics, 4(1), 2021.
  7. Edge-transitive direct products of graphs (with Cameron Crenshaw), under review.
  8. Graph exponentiation and neighborhood reconstruction, Discussiones Mathematicae Graph Theory,  41, 335-339, 2021.
  9. Sphere decompositions of hypercubes (with Paul Kainen), Art of Discrete and Applied Mathematics, 3(2): #P209, 2020.
  10. On 2-skeleta of platonic polytopes (with Paul Kainen), Bulletin of the Hellenic Mathematical Society, 62, 94-102, 2018.
  11. Cooperative Properties and Connected Sum (with Paul Kainen),  Cahiers de Topologie et Geometrie Differentielle Categoriques, 9(4) 388-404, 2018.
  12. Graph bases and diagram commutativity (with Paul Kainen),  Graphs and Combinatorics, 34(4), 523-534, 2018.
  13. Vertex-transitive direct products of graphs (with Wilfried Imrich), Electronic Journal of Combinatorics, 25(2), P2.10, 2018.
  14. Robust cycle bases do not exist for Kn,n when n > 8 (with Paul Kainen), Discrete Applied Mathematics, 235, 206-211, 2018.
  15. Neighborhood reconstruction and cancellation of graphs (with Cristina Mullican), Electronic Journal of Combinatorics, 24(2) 2017.
  16. Proper Connection of Direct Products of Graphs (with Dewey Taylor), Discussiones Mathematicae Graph Theory, 37(4) 1005-1013, 2017.
  17. Cycle bases of reduced powers of graphs (with Greg Smith), Ars Mathematica Contemporanea, 12(1): 183-202, 2017.
  18. Associativity and Non-Associativity of Some Hypergraph Products (with Marc Hellmuth, Lydia Ostermeier and Peter Stadler) Mathematics in Computer Science, 10(3) 403-408, 2016.
  19. Edge Transitive Products (with Wilfried Imrich and Sandi Klavzar), Journal of Algebraic Combinatorics, 43(4) 837-850, 2016.
  20. A prime factor theorem for bipartite graphs, (with Owen Puffenberger), European Journal of Combinatorics, 47, 123-140, 2015.
  21. Fast recognition of direct and strong products (with Wilfried Imrich), Ars Mathematica Contemporanea, 7: 487-497, 2014.
  22. Zero divisors among digraphs (with Heather Smith), Graphs and Combinatorics, 30(1), 171-181, 2014.
  23. Cancellation of digraphs over the direct product, European Journal of Combinatorics, 34: 846-858, 2013.
  24. On uniqueness of prime bipartite factors of graphs, Discrete Mathematics, 313: 1018-1027, 2013. (Here is my own version, identical but better formatted.)
  25. Frucht's theorem for the digraph factorial, Discussiones Mathematicae Graph Theory, 33: 329 -336, 2013.
  26. Notes on the independence number in the Cartesian product of graphs (with Ghidewon Abay-Asmerom, Craig Larson and Dewey Taylor), Discussiones Mathematicae Graph Theory, 31(1) 25-35, 2011.
  27. The inner power of a graph (with Neal Livesay), Ars Mathematica Contemporanea, 3: 193-199, 2010.
  28. An isomorphism theorem for digraphs (with Laura Culp), Australasian Journal of Combinatorics, 48: 205-211, 2010.
  29. Cancellation of direct products of digraphs (with Katherine Toman), Discussiones Mathematicae Graph Theory, 30(4): 575-590, 2010.
  30. A quasi cancellation property for the direct product, Discrete Mathematics, 310: 1691-1696, 2010.
  31. Direct product factorization of bipartite graphs with bipartition-reversing involutions (with Ghidewon Abay-Asmerom, Craig Larson and Dewey Taylor), SIAM Journal on Discrete Mathematics, 23(4): 2042-2052, 2010.
  32. On Cartesian skeletons of graphs (with Wilfried Imrich), Ars Mathematica Contemporanea, 2(2): 191-205, 2009.
  33. Proof of a conjecture concerning the direct product of bipartite graphs, European Journal of Combinatorics (Special Issue on Metric Graph Theory), 30: 1114-1118, 2009.
  34. Minimum cycle bases of direct products of graphs with cycles (with Zachary Bradshaw), Ars Mathematica Contemporanea, 2(1): 101-119, 2009.
  35. Factorial properties of graphs (with Ghidewon Abay-Asmerom and Dewey Taylor), Australasian Journal of Combinatorics, 44: 265-272, 2009.
  36. On direct product cancellation of graphs, Discrete Mathematics, 309: 2538-2543, 2009.
  37. Perfect r-codes in strong products of graphs (with Ghidewon Abay-Asmerom and Dewey Taylor), Bulletin of the ICA, 55: 66-72, 2009.
  38. Unique prime Cartesian Factorization of graphs over a finite field, Missouri Journal of Mathematical Sciences, 21(3): 149-154, 2009.
  39. Complementary composition of graphs (with Ghidewon Abay-Asmerom and Dewey Taylor), Congressus Numernatium, 189: 113-120, 2008.
  40. A cancellation property for the direct product of graphs, Discussiones Mathematicae Graph Theory, 28(1) 179-185, 2008.
  41. Total perfect codes in tensor products of graphs (with Ghidewon Abay-Asmerom and Dewey Taylor), Ars Combinatoria, 88: 129-234, 2008.
  42. Fractional graphs, Australasian Journal of Combinatorics, 39: 135-140, 2007.
  43. Minimum cycle bases of direct products of complete graphs, Information Processing Letters, 102: 214-218, 2007.
  44. Isomorphic components of direct products of bipartite graphs, Discussiones Mathematicae Graph Theory, 26(2): 231-248, 2006
  45. Minimum cycle bases of direct products of bipartite graphs, Australasian Journal of Combinatorics, 36: 13-26, 2006.
  46. Square contractions of graphs, Australasian Journal of Combinatorics, 36: 213-221, 2006.
  47. Centers of tensor products of graphs, (with Ghidewon Abay-Asmerom), Ars Combinatoria, 74: 201-211, 2005.
  48. Centers of n-fold tensor products of graphs, (with Sarah Bendall), Discussiones Mathematicae: Graph Theory, 24: 491-501, 2004.
  49. Equitable chromatic number of complete multipartite graphs, (with Dorothee Blum and Denise Grittner), Missouri Journal of Mathematical Sciences, 15: 75-81, 2003.
  50. A note on the complexity of computing cyclicity, Ars Combinatoria, 63:89-95, 2002.
  51. Cyclicity of graphs, Journal of Graph Theory, 32(2): 160-170, 1999.
  52. Circularity of planar graphs, Houston Journal of Mathematics, 25(2): 213-221, 1999.
  53. A homotopy equivalence between spaces of algebraic and holomorphic functions from an affine curve to a flag manifold, PhD thesis, UNC Chapel Hill, 1996.
TEACHING AND EXPOSITORY PUBLICATIONS, REVIEWS, etc:
  1. Book Review: Pop-Up Geometry: The Mathematics Behind Pop-Up Cards (by J. O'Rourke), Mathematical Intelligencer, 47:(1) 2025.
  2. Book Review: Pop-Up Geometry: The Mathematics Behind Pop-Up Cards (by J. O'Rourke), Math Horizons, 32:(1) 28-28, 2024
  3. Topological Envelopes, MAA FOCUS, December 2023/January 2024.
  4. Eulerian 2-complexes (with Paul Kainen), Mathematics Magazine, 97(1): 23-35, 2024.
  5. Report on Bridges Aalto 2022, Nexus Network Journal: Architecture and Mathematics,  25: 817-821, 2023.
  6. Closed surface envelopes,  Bridges Aalto 2022 Proceedings.
  7. A geometric view of complex trigonometric functions, College Mathematics Journal, 38(3): 210-217, May 2007.
  8. Alternating Series Convergence: A visual proof (with David Lyons), Teaching Mathematics and its Applications, 25(2): 58-60, 2006.
  9. An informal approach to formal inner products, Missouri Journal of Mathematical Sciences, 17: 153-160, 2005.
  10. Proof Without Words (with David Lyons), College Mathematics Journal, 32(1): p 72, 2005.
  11. A simple way to teach logarithms (with David Lyons), Mathematics Teacher, 88(5): 374-375, 1995.
SELECTED TALKS:
  1. The Hypercube Pop-Up Book, SIGMAA Special Session on Mathematics and the Arts, Joint Mathematics Meetings, Seattle, WA, January 10, 2025.
  2. The Hypercube Pop-Up Book, Maggie L Walker Governors School Chapter of Mu Alpha Theta, Richmond, VA, November 21, 2024
  3. Banquet talk: The Hypercube Pop-Up Book, MAA Section Meeting, Old Dominion University, Norfolk, VA, November 2, 2024.
  4. Plenary talk: Hypercube Models Through Genus Embeddings, Bridges Richmond, Richmond, VA, August 4, 2024.
  5. Mathematical pop-ups, Special Session on Mathematics and the Arts, Joint Mathematics Meetings, San Francisco,  CA, January 3, 2024.
  6. Topological envelopes, Special Session on Mathematics and the Arts, Joint Mathematics Meetings, Boston, MA, January 5, 2023.
  7. Efficient checking of diagram commutativity, Georgetown University Mathematics and Statistics Seminar, Washington, DC, November 4, 2022.
  8. Efficient checking of diagram commutativity, Maribor Graph Theory Conference, Maribor, Slovenia, September 15, 2022.
  9. Topological envelopes, Bridges 2022, Helsinki Finland, August 2, 2022.
  10. Plenary talk: Integrate THIS: The mathematics of planimeters,  RAMS Conference, Richmond, VA,  May 1, 2021.
  11. What makes a diagram commute?,  Summer Combo, University of Vermont, Burlington VT,  July 13, 2019.
  12. Edge transitive direct products of graphs, CanaDAM, Vancouver, BC, May 28, 2019.
  13. Not every graph has connected sum basis, Knots in Washington XLIV, The George Washington University, Washington DC, April 28, 2017.
  14. Not every graph has a robust cycle basis, LSU Graduate Mathematics Seminar, Baton Rouge, LA, March 9, 2017.
  15. Neighborhood reconstruction and cancellation of graphs, LSU Combinatorics Seminar, Baton Rouge, LA, March 8, 2017.
  16. Integrate THIS: The mathematics of planimeters, LSU Undergraduate Mathematics Colloquium, Baton Rouge, LA, March 7, 2017.
  17. Plenary talk: Integrate THIS: The mathematics of planimeters, SIAM Mathematics Awareness Conference, Old Dominion University, Norfolk, VA, April 9, 2016.
  18. The fourth dimension, Mu Alpha Theta Induction, Maggie L. Walker Governor's School, Richmond, VA, March 9, 2016.
  19. Edge transitive products, AMS Special Session on Graph Products, AMS-MAA Joint Meetings, Seattle, WA, January 9, 2016.
  20. Automorphism groups of bipartite direct products, 8th Slovenian International Conference on Graph Theory, Kranjska Gora, Slovenia, June 23, 2015.
  21. Non-Euclidean geometry and the visual arts, Pi Mu Epsilon Induction Ceremony, Lebanon Valley College, Annville, PA, April 18, 2015.
  22. Plenary Talk: Prime factorization of graphs over the direct product, (On-line speaker) Instructional Workshop in Graph Theory, Christ University, Bangalore, India, March, 2015.
  23. When can we cancel? Special Session: New Developments in Graph Theory, AMS Sectional meeting, Greensboro, NC, November, 2014.
  24. A prime factor theorem for bipartite graphs, Special Session: Recent Trends in Graph Theory, AMS-MAA Joint Meetings, Baltimore, MD, January, 2014.
  25. A do-it-yourself guide to on-demand publishing, Special Session on Open Source Textbooks, AMS-MAA Joint Meetings, Baltimore, MD, January, 2014.
  26. Did Caravaggio use a camera?, Mathematics Colloquium, Randolph-Macon College, October 25, 2013.
  27. Direct product cancellation of digraphs, Special Session: Recent Trends in Graph Theory, AMS-MAA Joint Meetings, Boston, MA, January 6, 2012.
  28. The factorial of a digraph, Mathematics Colloquium, University of Richmond, October 25, 2011.
  29. The factorial of a graph, Special Session: New Developments in Graph Theory, AMS Sectional meeting, Wake Forest University, September 25, 2011.
  30. Factoring direct products of graphs via Cartesian skeletons, Seminar: Montanuniversitat Leoben, Leoben, Austria, July 30, 2010.
  31. Prime factorization of bipartite graphs over the direct product, Seminar: University of Maribor, Maribor, Slovenia, July 30, 2010.
  32. Non-Euclidean geometry and the visual arts, Pi Mu Epsilon Induction Ceremony, Randolph-Macon College, April 30, 2010.
  33. An aspect of direct product cancellation of digraphs, Algebraic Graph Theory 2009, Dubrovnik, Croatia, June 1, 2009.
  34. Isomorphic components of direct products of bipartite graphs II, Joint Meetings of the MAA and AMS, Washington D.C., January 8, 2009.
  35. The anamorphic image, Randolph-Macon College, October 17, 2008.
  36. Direct product cancellation, 46th Midwest Graph Theory Conference, West Virginia University, Morgantown WV, April 26, 2008.
  37. A quasi cancellation property for the direct product, Sixth Slovenian International Conference on Graph Theory, Lake Bled, Slovenia, June 25, 2007.