Envelopes with Non-Standard Topologies

Richard Hammack

Here are some of my recent paper-and-glue envelope constructions. Each is an accurate (but flat) representation of a compact surface (such as a Klein bottle, torus projective plane, etc.) that can be sent through the mail. I see this work as being closely aligned with printmaking, with the postmarked stamp a stand-in for an edition number. The envelopes are the subject of a short paper (which you can find here) in the proceedings of Bridges 2022 at Aalto University, Finland. The YouTube video of my conference presentation (about 8 minutes) is here.

These artworks were exhibited at Bridges 2022 (Helsinki, Finland) and at the 2023 Joint Mathematics Meetings (Boston, MA). In Boston they won the prize for Best Photograph/Painting/Print.

         

Topologically, a standard (sealed) mailing envelope is a sphere. If the envelope has a window, then it is a sphere with a disk removed. These are mathematical objects. See what happens when strips are added to the window.


Projective Plane (with one hole), PDF-LaTeX, paper, glue, 2022

Here a folded strip has been added to a windowed envelope, creating one-sided surface with a hole (bounded by the thick black contour). This is a sphere with one cross-cap, also known as the projective plane.



Projective Plane (with three holes), PDF-LaTeX, paper, glue, 2022.

This envelope has a hexagonal window to which three interlocking folded strips are added. This creates three holes, bounded by the red, green and blue contours, respectively. Note that the red/green strip passes through the blue hole, the green/blue strip passes through the red hole, and the blue/red strip passes through the green hole. This is a flat variant of Boy's surface, and immersion of the projective plane in 3D space without pinchpoints. Indeed, widen the three strips and the three holes narrow into three slits through which the surface passes through itself.



Klein Bottle (with one holes), PDF-LaTeX, paper, glue, 2022.

This envelope is a sphere with two cross-caps (with a hole). Such a surface is commonly called a Klein bottle. See the example below for a more familiar representation of a Klein bottle.

         

Klein Bottle Envelope (with one hole)
, PDF-LaTeX, paper, glue, 2022.

This envelope is a flat Klein bottle. It is printed on a US legal-size sheet, cut, folded and glued into a flat tube with a window and a narrowed section that will becomes the neck of the bottle. It is then folded so that the neck tucks through the window and the two tube ends meet. Flaps on one end of the tube are glued to the other end.



            

Toroidal Envelope: front, back and selfie, PDF-LaTeX, paper, glue, 2022.

Having the topology of a genus-1 torus, this envelope is printed on a standard US letter-size sheet. The sheet is then folded along a horizontal axis, and again along a vertical axis. Flaps are glued to form a cylindrical pocket that is sealed by folding an upper flap to the inside. (For details, see my YouTube video.)