Bridges 2015
2-Manifold Sculptures
Carlo H. Séquin
EECS Computer Science Division
University of California, Berkeley
2-Manifold Sculptures
Endless Ribbon Dual Universe
Loop_785
Heptoroid
Max Bill
Charles Perry
Eva Hild
Brent Collins
stone
metal
ceramic
wood
Charles O. Perry

One of my heros!
A rich collection of topological sculptures!
Eva Hild
An even larger collection of ceramic creations & metal sculptures
“Tetra”, Waterfront Park, Louisville, KY
Charles Perry, 1999, bronze

Multiple views from the Web: Identify corresponding branches.
Modeling “Tetra” by Charles Perry (2)
 Crude
Assembling
labeled ribbons
Paper Models:
Un-twisted
tetrahedral frame
Twisted tetra frame
as in Perry’s “Tetra”
Modeling “Tetra” by Charles Perry (3)
Annotated
sculpture image
Metal-rings
Maquetteplus
of
scotch-tape
model
Perry’s “Tetra”
CAD model of
Perry’s “Tetra”
Continuum, Space Museum, Washington D.C.
Charles Perry, 1976, bronze
Continuum II
Singapore, 1986
“Continuum” by Charles Perry (2)

Multiple views
from the Web
(enhanced with
Photoshop)

Symmetry
becomes visible!
“Hollow” by Eva Hild, Varberg, 2006

A typical shot from the “front” side
“Hollow” by Eva Hild, Varberg (2)

A range of
front views …

+ Reflections!
“Hollow” by Eva Hild, Varberg (3)

Extracting and processing the reflected backside.
Parameterized Generators

For Creating Variations:

Ribbon-based:
Perry’s “Tetra”

Border-based:
Hild’s “Interruption”
Topological Analysis
Surface Classification Theorem:

All 2-manifolds embedded in Euclidean 3-space
can be characterized by 3 parameters:

Number of borders, b: # of 1D rim-lines;

Orientability, σ: single- / double- sided;

Genus, g, or: Euler Characteristic, χ,
specifying “connectivity” . . .
Determining the Number of Borders

Run along a rim-line
until you come back
to the starting point;

count the number of
separate loops.

Here, there are 4 borders:
Determining the Surface Orientability
A double-sided surface

Flood-fill paint the surface
without stepping across rim.

If whole surface is painted,
it is a single-sided surface
(“non-orientable”).

If only half is painted,
it is a two-sided surface
(“orientable”).
The other side can then be
painted a different color.
Determining Surface Orientability (2)
A shortcut:
If you can find a path to get
from “one side” to “the other”
without stepping across a rim,
it is a single-sided surface.
Determining the Genus of a 2-Manifold

The number of independent closed-loop cuts
that can be made on a surface, while leaving
all its pieces connected to one another.
Closed surfaces
Surfaces with borders
(handle-bodies)
(disks with punctures)
genus 0
genus 2
genus 4
All: genus 0
Determining the Euler Characteristic
Sometimes a simpler approach:

χ = V – E + F = Euler Characteristic

How many cuts to obtain a single connected disk?

Disk: χ = 1; every ribbon-join lowers χ by 1;
 thus “Tetra” ribbon frame: χ = –2
From this:

Genus = 2 – χ – b
for non-orientable surfaces;

Genus = (2 – χ – b)/2
for double-sided surfaces.
Volution_1
Carlo Séquin, 2003, Bronze

Double-sided (orientable)

Number of borders b = 1

Euler characteristic χ = –1
(2 cuts to produce a disk)

Genus g = (2 – χ – b)/2 = 1

Independent cutting lines: 1
Costa_in_Cube
Carlo Séquin, 2004, bronze

Double-sided (orientable)

Number of borders b = 3

Euler characteristic χ = –5

Genus g = (2 – χ – b)/2 = 2

Independent cutting lines: 2
“Endless Ribbon”
Max Bill, 1953, stone

Single-sided (non-orientable)

Number of borders b = 1

E.C. χ = 2 – 3 + 1 = 0

Genus g = 2 – χ – b = 1

Independent cutting lines: 1
“Möbius Shell”
Brent Collins, 1993, wood

Single-sided (non-orientable)

Number of borders b = 2 (Y,R)

Euler characteristic χ = –1

Genus g = 2 – χ – b = 1

Independent cutting lines: 1
blue or green – but not both,
they would intersect!
“Tetra”, Waterfront Park, Louisville, KY
Charles Perry, 1999, bronze

Double-sided (orientable)

Number of borders b = 4

Euler characteristic χ = –2

Genus g = (2 – χ – b)/2 = 0

It is a sphere with 4 punctures.

There are no closed-loop cuts
that leave this sculpture
connected !
D2d, Dartmouth College, Hanover, NH
Charles Perry, 1975, bronze

Double-sided (orientable)

Number of borders b = 4

Euler characteristic χ = –2

Genus g = (2 – χ – b)/2 = 0

It is a sphere with 4 punctures.
Modification of Perry’s “Tetra” Sculpture

Using my generator for tetrahedral ribbon frames,
individually adjusting the twist of all six ribbons:
Untwisted tetra frame – Emulating Perry’s “Tetra” and “D2d”
4 ribbons have a ±360 twist
“Tetra_2T”
Modification of Perry’s Tetra Sculpture

This modified version has only TWO twisted tetra-edges.

This does not change the number of borders,
but it leaves them only pair-wise interlinked.
Schematic of modification

Double-sided (orientable)

Number of borders b = 4

Euler characteristic χ = –2

Genus g = (2 – χ – b)/2 = 0

A sphere with 4 punctures.
“Tetra_6T”
Modification of Perry’s Tetra Sculpture

Here all six tetra-edges twist through 360.

Surface classification does not change,
but all border-pairs are now interlinked.
Original -- Modified
“Tetra_4M”
Modification of Perry’s Tetra Sculpture

The four twisted tetra-edges rotate through only 180.

3D-Print, painted
This keeps the surface double-sided,
but only 2 (different) borders.
Original -- Modified

Double-sided (orientable)

Number of borders b = 2

Euler characteristic χ = –2

Genus g = (2 – χ – b)/2 = 1

It is a torus with 2 punctures.
“Tetra_2M”
Modification of Perry’s Tetra Sculpture

Only TWO tetra-edges are twisted through only 180.

This now makes it single-sided!
This now has only two borders (both identical).
Original -- Modified

Single-sided (non-orientable)

Number of borders b = 2

Euler characteristic χ = –2

Genus g = 2 – χ – b = 2

Klein-bottle with 2 punctures
“Tetra_6M”
Modification of Perry’s Tetra Sculpture

All SIX tetra-edges are twisted through –180.

This also makes it single-sided!

This one has three identical borders,
Forming a Borromean link !

Single-sided

Number of borders b = 3

Euler characteristic χ = –2

Genus g = 2 – χ – b = 1
“Tetra_3M”
Modification of Perry’s Tetra Sculpture

3 tetra-edges are twisted through 180.

Single-sided

Number of borders b = 1

Euler characteristic χ = –2

Genus g = 2 – χ – b = 3
Results of “Tetra” Modifications
NAME
Tetra_0T
Perry_4T
Tetra_2T
Tetra_6T
Tetra_4M
Tetra_2M
Tetra_6M
Tetra_1M
Tetra_3Mo
Tetra_3Mz
Tetra_3My
Tetra_5M
sigma
2
2
2
2
2
1
1
1
1
1
1
1
borders
4
4
4
4
2
2
3
2+1
1
1
1+1
1+1
E.C.
-2
-2
-2
-2
-2
-2
-2
-2
-2
-2
-2
-2
genus
0
0
0
0
1
2
1
1
3
3
2
2
b-linking
each: 0
each: 2
each: 1
each: 3
0, 0
0, 0
Borrom.?
0, 0, 0
n.a.
n.a.
0, 0
0, 0
“Dual Universe”, Shell Plaza, Singapore
Charles Perry, 1994, bronze

Double-sided (orientable)

Number of borders b = 2

Euler characteristic χ = –2

Genus g = (2 – X – b)/2 = 1

It is a torus with 2 punctures.

Still four 3-valent junctions;
but no longer a tetra-frame:
2 pairs with 2 connections!
“Duality”, Pugh Residence, Greenwich, CT
Charles Perry, 1986, bronze

Double-sided (orientable)

Number of borders b = 2

Euler characteristic χ = –2

Genus g = (2 – χ – b)/2 = 1

It is a torus with 2 punctures.

There is one closed-loop cut
that leaves this sculpture
connected !
“Duality_2”, Barnett Center, Jacksonville, FL
Charles Perry, 1990, bronze

Double-sided (orientable)

Number of borders b = 4

Euler characteristic χ = –2

Genus g = (2 – χ – b)/2 = 0

It is a sphere with 4 punctures.
“Continuum”, Space Museum, Washington D.C.
Charles Perry, 1976, bronze

Single-sided (non-orientable)

Number of borders b = 1

Euler characteristic χ = –5

Genus g = 2 – χ – b = 6

Independent cutting lines: 6
“Interruption”
Eva Hild, 2002, ceramic

Double-sided (orientable)

Number of borders b = 2

Euler characteristic χ = –2

Genus g = 1 (after closure)

Torus with 2 punctures

Independent cutting line:
“Hollow” by Eva Hild, Varberg, (2006)

Double-sided (orientable)

Number of borders b = 1

Genus g = 2 (after closure)

2-hole torus with 1 puncture
“Hyperbolic Hexagon”
Brent Collins, 1996, wood

Double-sided (orientable)

Number of borders b = 4

Euler characteristic χ = –6

Genus g = (2 – χ – b)/2 = 0
“Heptoroid”
Collins & Séquin, 1997, wood

Single-sided (non-orientable)

Number of borders b = 1

Euler characteristic χ = –21

Genus g = 2 – χ – b = 22

Independent cutting lines: 22
Minimal Trefoils
Collins & Séquin, 1997, wood & bronze

Single-sided (non-orientable)

Number of borders b = 1

Euler characteristic χ = –3

Genus g = 2 – χ – b = 4

It is difficult to place the
4 independent cutting lines!
More of the same …
Min. Trefoil
Sculpture Gen.

Single-sided (non-orientable)

Number of borders b = 1

Euler characteristic χ = –3

Genus g = 2 – χ – b = 4
2 Klein bottles
R. Roelofs
“Connected sum of two Klein bottles with a single puncture”
Beyond Sculptures . . .

No reason to limit this kind of analysis to sculptures!

May also be applied to vases or baskets …
Iranian 4-lobed Vase,
ca. 1100-1300

Double-sided (orientable)

Number of borders b = 1

Genus g = 3

E.C. χ = 2 – b – 2g = –5
Bamboo Basket
“Galaxy” (Sejun)
by Honda Shoryu
2001

Double-sided (orientable)

Number of borders b = 4

Sphere with 4 punctures

Genus g = 0

E.C. χ = 2 – b – 2g = –2
More of the same …
H. Shoryu
Perry: “D2d”

Double-sided (orientable)

Number of borders b = 4

Euler characteristic χ = –2

Genus g = 0
{sphere}
Sculpture Gen. 3-hole button
Summary

The Surface Classification Theorem
has been applied to various 2-manifold sculptures.

Extracting the 3 crucial parameters, b, σ, χg,
is not always easy (~ detective game).

But once you have found those defining parameters,
they yield a key for a deeper understanding, and
they help to retain a mental image of a sculpture.
Conclusions

Topology is fun!

It is quite intriguing to find these relationships
between very different-looking sculptures.

But this is only a first, very coarse classification.
(Next, one could look at the linking of the borders.)

The beauty of a sculpture does not primarily come
from its topology, but from its geometry.

That is why we need artists like Charles Perry and
Eva Hild who can cast a particular topology into a
beautiful geometry.
Questions ?
Charles Perry
Eva Hild
SPARE
Things you understand . . .
. . . are easier to memorize
Operational amplifier
Artistic circuit
Tripartite Unity
Max Bill, 1949, stainless steel

Single-sided (non-orientable)

Number of borders b = 1

Euler characteristic χ = –2

Genus g = 2 – χ – b = 3
More of the same …
Max Bill
Dyck Surf.

Single-sided (non-orientable)

Number of borders b = 1

Euler characteristic χ = –2

Genus g = 2 – χ – b = 3
Sculpture Gen.
Twisted “Tetra”