Delivered at Bridges Aalto 2022: Dessins d'enfants, famous in number theory, are a special sort of drawing made on a closed surface like the sphere. For our purposes, a dessin shows how to seamlessly tile a closed surface with copies of Earth's surface. These tiled surfaces can fold up into a single copy of Earth's surface. Includes instructions for weaving a cubical paper basket of this kind. See accompanying paper at archive.bridgesmathart.org

1. Bridges Aalto 2022, 1-5 August 2022
James Mallos, Sculptor
Washington, DC, USA
jbmallos@gmail.com
Geoweaving:
Fold-Up Baskets from Dessins d Enfants

2. What is a dessin d enfants?

3. Gareth Jones “Dessins d’Enfants”
https://youtu.be/Bkva3x8wgZU
Number theory made dessins famous̶see Gareth Jones
video̶but here we ll stay close to drawing and weaving.

4. A dessin is a special sort of drawing made on a
closed, oriented surface.

5. Some closed surfaces cannot be oriented, but those that
can have 2 sides, and we need to agree which side of the
surface we are standing on. Only then we can say words
like left and counterclockwise with the usual assurance.
CCW

6. If we have an illustration of the surface, the
convention that we illustrate surfaces like they are
terrestrial planets we are about to set foot on, may
serve as a tacit orientation of the surface.
CW

7. The rules for drawing a dessin are particularly easy
when the oriented surface happens to be a sphere.
Then the drawing just needs to be connected.

8. “...any child’s drawing scrawled on a
bit of paper (at least if the drawing is
made without lifting the pencil)
gives a perfectly explicit example.”
—Alexander Grothendieck
In the words of Alexander Grothendieck, the originator
of the theory, drawn without lifting the pencil.

9. Or, just draw any way you like...

10. ...and then add lines connecting the others.

11. On a surface more complex than the sphere, being
connected is not enough. The drawing must cut the
surface into topologically trivial pieces̶on a sphere
that s just happens to always be the case.
Oops!

12. A bit of paper makes a perfectly good substitute for
a sphere if we don t need to draw on the backside
(and we don t forget there is a backside.).

13. How do we turn a
connected drawing on the
sphere into a proper
dessin?

14. 1. Add black dots where lines end or cross.

15. 2. Add a white dot in the middle of each edge.

16. A dessin, voilà!

17. Notice that a dessin may be said to have an outer
face. The outer face here is bounded by 6 edges.

18. When we say a dessin we actually refer to a topological
equivalence class. For example, a dessin is unchanged if
re-drawn to show us a di
ff
erent face as its outer face.
=

19. The theory of dessin d enfants is the deeper, more symmetrical
theory that lies behind graph theory.
Dessins are everywhere.
Lando and Zvonkin Graphs on Surfaces and Their Applications (2004)

20. The Adams Tile

21. In 1929, Oscar Sherman Adams, published a new
map projection he dubbed World in a Square II.

22. Adams projection makes some places
look big, and others look small...

23. But if you zoom in close enough, everything
looks true to life. This property is called
conformality.

24. Conformality fails in a big way at the corners of
Adams square. 360 or 180 degrees (on Earth) are
squeezed into the 90-degree corners of the map.
360°-> 90°
360°-> 90°
180°-> 90° 180°-> 90°

25. Zooming in on these points is
not going to
fi
x this.
360°

26. The really remarkable property of Adams projection is
that it tiles into a map of a periodic Earth.

27. This map is classed as meromorphic because it is conformal
except at isolated points (critical points) where tile corners meet.

28. There might be any number of critical points in an Adams
tiling, but they image just 3 points (critical values) back
home on Earth: South Pole, North Pole, the Mid-Paci
fi
c Point.

29. Magellan
How
many times
should I go around
Antarctica?
What would it be like to live on a periodic Earth?

30. In order to cover a sphere with a single Adams tile̶a.k.a., modeling
a normal Earth̶the Adams tile must fold along its Prime Meridian.
(That s the only way its edges can coincide properly paired.)

31. OK, let s get back to dessins.

32. Each dessin shows a way Adams tiles can
be arranged to tile its surface.

33. Each edge (oriented black towards white) shows how a tile s
Prime Meridian (oriented south towards north) should be placed.

34. Antarctica -> Black vertices
Arctic Ocean -> White vertices
Paci
fi
c -> Face centers (Pink)

35. Don t we need stretchy
tiles to conform to our
drawing?

36. No, the surface we drew
on is already stretchy
(topological.) We can
imagine that surface
conforming to whatever
surface the joined Adams
tiles form.

37. Let s try taping together some Adams
tiles as directed by our dessin.

38. It s got 4 edges, so we need 4 tiles.

39. The vertex at the tail is degree 1, so that tile must
join to itself forming a normal Southern Hemisphere.

40. The next vertex is degree 2, so 2 tiles join
there, with matching corner colors .

41. The next vertex is degree 3, so 2 new tiles
must join there. Here s the one on this side.

42. And here s the one on the other side.

43. The last vertex is degree 2, so we need to join 2 tiles there̶but
they are already in the work̶so we just need to apply tape.

44. It s wise to check that the faces ended up with the right number
of sides̶yep, 6 pink corners for that pesky outer face.

45. It s not hard to
fi
gure out a way to fold the surface
into one triangle displaying 1 copy of the globe.

46. If you need convincing that this is really a topological
sphere, tape the joins up completely except for about a
1/2 cm opening̶and pu
ff
it up with a straw.

47. How can we weave this as
a basket?

48. The weaving elements are straight strips of
Adams tiles that always cross dessin edges at
45 degrees.

49. After that, they cross an adjacent edge on the
same face.

50. We can sketch the paths like that on the dessin by
fi
rst
putting an X in the middle of each edge.

51. Then link the X s with bouncing ball trajectories
that tour around the interior of each face.

52. This 4-regular graph, called the medial, contains
the paths of all the weaving elements.

53. Explore individual paths by tracing in colors.

54. Tracing gives you the number tiles in each element (1
per crossing) and helps you choose a place to start.

55. If you have a model of the surface, lines of masking
tape give the same information.

56. In the 1990 s Heinz Strobl invented a type of weaving called
knotology. Geoweaving elements have the same crease
pattern, but less symmetry due to their printed pattern.

57. Knotology is more versatile because black
and white vertices need not be distinguished
(e.g., paint both gray.)

58. Going beyond drawings
as a source of dessins

59. A graph is said to be bipartite if its vertices can be
alternately colored black or white such that no edge joins
vertices of the same color. (Most graphs are not bipartite̶ a
triangle or an odd cycle of any size, spoils the broth.)
?

60. A bipartite graph drawing has 2 possible
bicolorings.

61. But, in a few cases, the alternate
bicoloring is self-equivalent̶there s
really only 1 bicoloring in such a case.

62. Graphs properly drawn on the sphere,
when considered up to topological
equivalence, are known as planar maps.

63. All the possible dessins on the sphere
coincide with the bicolored planar maps:

64. Some of these bicolorings cannot be made by
inserting white vertices into a drawing, because
that method only makes white vertices of degree 2.

65. The better strategy is to search the bipartite
planar maps, testing for self-equivalence.

66. Bipartite planar maps and their bicolorings are
enumerated by these OEIS integer sequences.
≈ 2
Nearly all bipartite
planar maps have
2 bicolorings.

67. So, what s the big deal
about being bipartite?

68. Bicolored maps model a generalization of
graph theory where an edge can connect
any number of vertices̶not just 2.

69. White vertices represent
these edges (hyperedges.)

70. Think of these white vertices
as n-legged octopuses...

71. ...that use their n-legs to link
to black vertices.

72. Why do dessins fold up into a single triangle?

73. Despite our child-like freedom to draw any connected
drawing on the sphere, our drawings inevitably induce a
topological partition of the sphere.

74. Let s see that by translating our (Walsh) representation
of dessins into the Canonical Triangulation.

75. First, insert a pink vertex in the middle of each face.

76. Add edges radiating out from each pink vertex to the
black and white vertices that bound its particular face.

77. The sphere has now been partitioned into
topological triangles.

78. Each triangle has: 1 black, 1 white, & 1 pink vertex.

79. We re going to separate the triangles and
temporarily forget how to put them back together.

80. But
fi
rst, let s make this remark: The pairs of edges we need
to identify to put the surface back together are coincident.

81. When we have cut the triangles apart (and stretched them a bit)
we re going to
fi
nd a new con
fi
guration that allows us to make the
same remark.

82. Now, we ve cut the triangles apart̶and temporarily forgotten how
to put them back together. Being mainly concerned with topology,
we ve gone ahead and stretched them into 45-90-45 triangles of
standard size̶the 90-degree angle at the pink vertex.

83. There are 2 classes of triangles (i.e., the cyclical order
of the colors Black, White, Pink is either CCW or CW),
so we can organize them into 2 stacks with matched
corner colors.
CCW CW

84. We can do one better if we
fl
ip over one of the stacks
(say, the CW stack.) Then all the triangles can be
stored in one stack with matched corner colors.

85. Our mathematical triangles all have zero
thickness, so they are not really stacked
but actually coincident.

86. Now we can repeat that remark: The
pairs of edges we need to identify to put
the surface back together are coincident.

87. Suddenly, we remember the pairs of edges we wanted
to identify to put the surface back together. Nothing
needs move, one snap of the
fi
ngers, presto!..

88. ...the surface is back together, whole, but in a completely
folded con
fi
guration. This is a mathematical (phantom)
folding, so there s a chance that no corresponding
physical (self-avoiding) folding would exist.

89. Let s now start unfolding all the folds at once by bellying
downward the turned-over, CW triangles, and bellying
upward the right-side-up, CCW triangles, making a
triangular pillow.

90. Let s keep in
fl
ating the triangular pillow
until it becomes a sphere. Each triangle
has stretched into a hemisphere.

91. There are 4 copies of each hemisphere (our
bicolored map had 4 edges), so a generic
point on the sphere is covered 4 times over.

92. But what happens at the vertices? Our bicolored map
happened to have 2 vertices of each color, so the sphere
must be covered just twice at those isolated points.
(Yikes!, how s that even possible?)

93. This strange wrapping of surface wrapped
on a sphere is called a rami
fi
ed covering
of the sphere.

94. Instructions to Geoweave a
Fold-Up Basket

95. Print out the weaving elements
on sti
ff
paper and cut them out.

96. Make the 90-degree folds (indicated by the
dashed lines in the margins) by pressing
against a sharp straight-edge.

97. Fully crease those folds.

98. Fully crease them in the
other direction.

99. Make the 45-degree folds indicated
by the dashed lines in the margins.

100. Fully crease those folds by making an
accordion fold; then re-fold it the other way.

101. Trim o
ff
the light-colored margins,
allowing the weaving some looseness.

102. Notice there are 2 types: the middle fold
displays either the North or the South Paci
fi
c.

103. Alternate the 2 types to weave a doubled
version of the Mid-Paci
fi
c Point at their
center folds.

104. Hold the crossing together while you
fl
ip
it over. Adjust it by aligning the creases.

105. Lessen the stickiness of a piece of removable tape by
sticking it to your skin multiple times; then fold back a
corner and apply it to the crossing to keep it in alignment.

106. Turn the work over (you now see its base.)
Fold downward to form the edges of the base.

107. Flip the work over; keep an over-and-under rhythm as you
form the vertical corners with diagonally crossing elements,
using paper clips to temporarily hold elements in place.

108. When all four walls are clipped, remove the tape.

109. Splice the weaving elements at the top with clear tape
after you ve maintained the over-and-under pattern.

110. James Mallos, Sculptor
Washington, DC, USA
jbmallos@gmail.com
Kiitos!