The Flat Klein Bottle Rendered in Curved-Crease Origami
TThe Flat Klein Bottle Rendered in Curved-Crease Origami
Stepan PaulNorth Carolina State University; [email protected]
Abstract
We introduce a simple and concrete way of visualizing in three dimensions a “flat” Klein bottle—one whose localintrinsic geometry is the same as that of a flat plane—which preserves most its topological and geometric structure.Concretely, the flatness property means that a small patch of the surface around any point can be flattened to apatch of the plane without stretching or compressing. Thus we can use the medium of curved-crease origami withinelastic film to make a model which, except for its self-intersections, necessarily has the flatness property, even alongits folded edges. As such, the sculpture presented here illustrates both the flatness, and, through its coloring, thenon-orientability of a Klein bottle.
Introduction
We present here a sculpture of a flat Klein bottle meant to illustrates both its topological and geometricproperties.
Figure 1:
A curved-crease origami sculpture of a flat Klein bottle made from printed transparency film.
Here the term “flat” is used to mean that the surface has the same local intrinsic geometry as a flatplane at every point. I elaborate on this property and its significance in the next section, but intuitively, itmeans that a patch of the surface around any point can be deformed into a flat plane without stretching orcompressing—that is, without distorting distances. a r X i v : . [ m a t h . HO ] J a n n the present sculpture, the uniform grid pattern and origami medium are intended to illustrate theflatness of Klein bottle, and the coloring of the grid is intended to illustrate its non-orientability.We think of the underlying mathematical figure ∇ , which consists of three circular cylinders of equalradius centered around the axes of an equilateral triangle, represented by the sculpture as the image of theabstract flat Klein bottle under a map 𝜑 into euclidean three-space with the following properties.I. 𝜑 is continuous everywhere, and 𝜑 is smooth except along a finite set of curves,II. 𝜑 is locally geometry-preserving (a local isometry) except at a finite set of points, andIII. 𝜑 is one-to-one, except along a finite set of curves.It is not possible to embed a Klein bottle into euclidean three-space, and thus neither the topology northe geometry can be completely preserved. However, this set of properties for 𝜑 can be thought of as anapproximation to topology- and geometry-preservation. To the author’s knowledge, this is the first explicitexample of a map from the flat Klein bottle into euclidean three-space satisfying these conditions, and perhapsthe first attempt to concretely visualize flat geometry on the Klein bottle in three dimensions.In what follows, I start with an overview of the mathematical concepts at play and prove that the claimedproperties of 𝜑 follow from the theory of curved-crease origami. I then elaborate on the construction andintention of the sculpture itself, focusing on how it self-evidentially illustrates important properties of theKlein bottle. Finally, I then give some mathematical and artistic context by exploring the relationship of thepresent illustration to related works. Mathematical Overview
In this section, we explain in a bit more detail what is meant by “flatness”, how it relates to the Klein bottleand torus specifically, and how the theory of curved-crease origami is applied in constructing the figure ∇ . Flatness
As we mention in the introduction, a surface is said to be flat if it has the same local intrinsic geometry asthe euclidean plane. This means that for any point on the surface, a piece of the euclidean plane can map isometrically onto a neighborhood of the point (i.e. the map is such that the distance between two points in theplane is the same as the distance between their images along the surface ). Examples of flat surfaces includethe cylinder and the non-smooth surface consisting of two half-planes meeting at any nonzero dihedral anglealong a line, as illustrated in Figure 2. (a) (b) (c)
Figure 2:
Examples of flat surfaces: A piece of the euclidean plane (decorated with an image) mapsisometrically (with the image undistorted) onto a cylinder (b) and a non-smooth surface (c). y Gauss’
Theorema Egregium , a smooth flat surface can be characterized extrinsically by the fact thatits Gaussian curvature is everywhere zero, but as in Figure 2c, a flat surface need not be smooth.In contrast, a spherical surface is not flat since it cannot be flattened without distorting distances.Likewise, the surface of a cube and the surface of a solid cylinder are not flat even though their faces are;neither a neighborhood of a vertex of the cube, nor a neighborhood of a point along the edge of the cylindercan flattened without stretching or ripping. (a) (b) (c)
Figure 3:
Examples of non-flat surfaces: A piece of the euclidean plane cannot be mapped (a) to a sphere,(b) onto a neighborhood of a vertex of a cube, nor (c) onto a neighborhood of a point along theedge of the surface of the solid cylinder without distorting distances.
In a way, the obstruction to flatness for all three of these examples is the same; their global topology—ineach case, that of a sphere—restricts the geometry. Indeed, it is a consequence of the Gauss-Bonnet Theoremthat any flat compact surface (without boundary) is topologically either a torus or Klein bottle.As we discuss in the section on the mathematical and artistic context of this work, the flat torus has beenillustrated in several previous works, and the present sculpture seeks to do the same for the flat Klein bottle.
The Flat Klein Bottle
A flat Klein bottle can be constructed by identifying the edges of a euclidean rectangle as show in Figure 4.
Figure 4:
Edge identifications on a euclidean rectangle to form a flat Klein bottle.
A locally isometric map on the rectangle that respects the edge identifications can then be interpreted asa local isometry on the Klein bottle.
Theory of Curved-Crease Origami
In constructing the map 𝜑 and its image ∇ , we apply the theory of curved-crease origami found, for example in[2]. Denote by E and E the euclidean plane and three-space respectively. Let 𝛼 be a curve in E —thoughtof as a curve drawn on a flat piece of paper—and let 𝑓 be a smooth local isometry of a neighborhood of 𝛼 into E —thought of as a particular positioning of the paper in E , possibly bending (but not folding) thepaper.he theory then says that if for every point 𝑥 along 𝛼 , the curvature of 𝑓 ( 𝛼 ) at 𝑓 ( 𝑥 ) is strictly greaterthan the curvature of 𝛼 at 𝑥 , then there exists a unique smooth local isometry 𝑓 that is different from 𝑓 butwhich also maps a neighborhood of 𝛼 into E and which agrees with 𝑓 along 𝛼 . We interpret this to meanthat there is exactly one other way to position the sheet of paper that places the drawn curve 𝛼 in the samelocation that 𝑓 did. Furthermore, at points where the curvature of 𝛼 is non-vanishing, the piecewise-definedfunction 𝑓 defined to be 𝑓 on one side of 𝛼 and 𝑓 on the other side is also a local isometry.We apply this theory to the example of the elliptical intersection of a circular cylinder and a plane. Inparticular, if Ψ is a circular cylinder of radius 𝑟 and Π is a plane meeting the axis of Ψ at an angle of0 < 𝜃 < 𝜋 , then one can choose a coordinate system so that the map 𝑓 ( 𝑢, 𝑣 ) = (cid:16) 𝑟 sin 𝑢𝑟 , 𝑣, 𝑟 cos 𝑢𝑟 (cid:17) (1)maps (locally isometrically) onto Ψ and so that Π has equations 𝑦 = 𝜏𝑥 where 𝜏 = tan 𝜃 .The pre-image of the the elliptical intersection 𝜖 is the curve 𝛼 in R defined by 𝑣 = 𝜏𝑟 sin 𝑢𝑟 . Applyingthe theorem locally, 𝑓 | 𝛼 extends to 𝑓 and exactly one other smooth local isometry 𝑓 , and we can immediatelyidentify 𝑓 as the reflection 𝜌 ◦ 𝑓 , where 𝜌 is reflection across Π (since 𝜌 fixes 𝜖 ), and thus the piecewisefunction 𝑓 ( 𝑢, 𝑣 ) = (cid:40) 𝑓 ( 𝑢, 𝑣 ) 𝑣 ≤ 𝜏 sin 𝑢𝑟 𝑓 ( 𝑢, 𝑣 ) 𝑣 ≥ 𝜏 sin 𝑢𝑟 (2)is a local isometry away from the inflection points of 𝛼 . 𝛼 (a) (b) (c) Figure 5:
The 𝑢𝑣 -plane in (a) is mapped by 𝑓 onto the solid cylinder in (b). The dashed curve 𝛼 in (a)maps to the dashed intersection of the solid cylinder with a plane in (b). The image of 𝑓 is shownas a transparent cylinder in (b). The image of the piecewise function 𝑓 is shown in (c), and theinflection points of 𝛼 and their images are highlighted, and the semi-ellipse where the twocylinders intersect and its pre-image are shown as dotted curves in (c) and (a) respectively. One further observation to make here is that the “handedness” of the cylinder parameterizations 𝑓 and 𝑓 are different in the following sense. If, while one points the thumb of one’s right hand in the directionof increase in the axial variable ( 𝑣 ) one’s fingers curl in the direction of increase in the angular variable ( 𝑢 ),we say that the cylinder parameterization is “right-handed”; otherwise it is left-handed. Mathematically,the normal vector resulting from the cross product of the partial derivatives 𝜕 𝑓𝜕𝑢 × 𝜕 𝑓𝜕𝑣 points outward for aright-handed parameterization and inward for a left-handed parameterization; the reversal results from thefact that 𝑓 is a reflection of 𝑓 . onstructing ∇ and 𝜑 One can now imagine patching together copies of this basic construction to build the figure ∇ as the image ofa local isometry from a subset of E . We’ll show that by patching appropriately, this subset can be a rectanglewith the necessary edge identifications from Figure 4.Indeed, by setting 𝜃 = 𝜋 and restricting the constructed map 𝑓 to the rectangle 𝐷 where − 𝜋𝑟 ≤ 𝑢 ≤ 𝜋𝑟 and − 𝑠 ≤ 𝑣 ≤ 𝑠 , the image of 𝑓 is one-third of the figure ∇ . Changing coordinates appropriately, one canconstruct a map 𝜑 from the rectangle consisting of three stacked copies of 𝐷 to three copies of the image of 𝑓 glued together to form ∇ . 𝐷 𝐷 𝐷 − (a) (b) Figure 6:
Gluing copies of the basic construction. The rectangular domain (a) consists of stacked copies of 𝐷 ; the creases are shown as dotted lines and the oriented gluing edges are marked with arrows.Glued appropriately, we obtain the figure ∇ in (b). We note that it is clear from the formula (1) that 𝑓 respects the identification of the vertical edges, andtherefore 𝜑 does as well. By construction, the horizontal edges both get isometrically mapped to the samecircle, and since the handedness of the cylinder parameterization is switched an odd number of times, theorientation of these maps must be opposite one another, as required. In this way, 𝜑 only fails to be injectivealong the three semi-elliptical pairwise intersections of the cylinders highlighted in figure 5c. The rest of theclaimed properties of 𝜑 from the Introduction follow from the basic construction: appropriate gluing ensuresthat 𝜑 is everywhere continuous, 𝜑 is smooth except along the three copies of 𝛼 (which map to the foldedellipses), and 𝜑 is a local isometry except at each of the two inflection points of each copy of 𝛼 .We note that similarly taking the union of appropriate pieces cylinders of equal radii centered aroundthe edges of any 𝑛 -sided polygon in E will result in a figure that is the image of torus if 𝑛 is even or a Kleinbottle if 𝑛 is odd. Illustration Through Sculpture
The aim of the present sculpture is largely a pedagogical one: to make concrete the consequences of conceptstaught in courses on topology and geometry. The Klein bottle is a familiar object to mathematicians in theabstract, and while sculptures and drawings like the one in Figure 10a help one understand its topology, they a) (b)
Figure 7:
By appropriately gluing 𝑛 copies of the basic construction, one obtains (a) an image of the toruswhen 𝑛 is even or (b) a Klein bottle when 𝑛 is odd. do not attempt to illustrate its admission of flat geometry.I made the sculpture from clear, printed transparency film by cutting, creasing, and taping. While thecreases are meant to be seen, the cuts and tape are hidden, giving the illusion of physical continuity toillustrate the underlying mathematical continuity. The grid pattern and transparent medium are further meantto aide the viewer in willfully looking past the self-intersections and thereby focus on the properties of thesculpture inherited from the abstract Klein bottle. In particular, the cylindrical pieces are meant to appear topass through each other—their semi-elliptical intersection (achieved by cutting one of the cylinders to allowthe other to pass through) easily ignored. (a) (b) Figure 8:
Details of a self-intersection (a) and folded ridge (b) on the sculpture.
Consideration of these self-intersections suspended, the sculpture apparently represents a compactgenus-one non-orientable surface (the non-orientability emphasized by the coloring of the grid pattern). Aviewer with some background in topology will quickly identify this as the Klein bottle, even if this particularmanifestation is unfamiliar.A viewer may then recognize that the construction from inelastic transparency film demonstrates the flatgeometry of the surface. As already noted, compact surfaces with topology different from the Klein bottleand torus do not admit flat geometry. The use of origami, including curved-crease origami, to model surfaceswith flat metrics has been studied before [4, 2], and in this way, the medium itself illustrates the flat metric onhe Klein bottle. The grid pattern is designed to further emphasize this flatness. As noted earlier, a “texture”on a flat plane can be mapped undistorted onto a flat surface, even across folds and vertices, as it is in thepresent sculpture.
Mathematical and Artistic Context
The context of the present sculpture of the flat Klein bottle and its mathematical underpinnings is bestunderstood by exploring the more well-developed set of works regarding the flat torus.As we mentioned earlier, any flat compact surface necessarily has the global topology of either a Kleinbottle or torus. Unlike the Klein bottle, the torus may be smoothly embedded in euclidean three-space.However, no smooth embedding of the torus can also preserve the geometry of a flat torus; that is, it cannotbe both smoothly and isometrically embedded (notice the necessary distortion in Figure 9a).However, explicit examples of continuous piecewise-linear isometries of the flat torus into euclideanthree-space do exist, as illustrated by Segerman using 3D printing in [8] and Málaga and Lelièvre in [6] usingorigami. (a) (b) (c)
Figure 9:
Embeddings of the flat torus: (a) A smooth embedding of the torus cannot be flat. (b) An origamisculpture of a piecewise-linear flat torus [6]. (c) A 3D printed sculpture of a continuouslydifferentiable flat torus [3].
In fact, Nash [7] and Kuiper [5] proved theorems in the 1950s which show that there exists a continuouslydifferentiable (but not twice-differentiable) isometric embedding of the flat torus into euclidean three-space,although its image is necessarily a fractal. An explicit example of such an embedding was not found until2012 by Borrelli et al in [1], and a 3D printed model, pictured in Figure 9c, of the embedded flat torus waspresented by Henocque and Marin in [3].One can view the present sculpture as filling in a piece of a parallel and mostly untold story about theKlein bottle. The Klein bottle cannot be globally embedded in euclidean three-space, but it can be smoothlylocally embedded as in Figure 10a. The results of Nash and Kuiper show that smoothness can be tradedfor local isometry and continuous differentiability, but no explicit example of such an isometry has beenexhibited. We can think of the map 𝜑 constructed here as satisfying a weaker of the set of properties thanthose guaranteed by Nash and Kuiper. Thus the construction of ∇ and its sculpture are presented as a firststep towards developing illustrations of the flat Klein bottle that parallel those exhibited for the flat torus. Summary and Conclusions
While the existence of a flat Klein bottle in the abstract is well-known to mathematicians, I hope that byexhibiting its image in euclidean three-space with much of the topology and geometry still intact will helpmake this fact feel more concrete, especially to those encountering such ideas for the first time. a) (b)
Figure 10:
Images of the Klein bottle: (a) A smooth local embedding of the Klein bottle cannot be flat. (b)A sculpture of ∇ . Acknowledgements
This material is based upon work supported by the National Science Foundation under Grant No. DMS-1439786 and the Alfred P. Sloan Foundation award G-2019-11406 while the author was in residence atthe Institute for Computational and Experimental Research in Mathematics in Providence, RI, during theIllustrating Mathematics program. Along the many other artists and mathematicians I met and worked withat ICERM whose ideas worked their way into this project, I wish to thank John Edmark for introducing meto curved-crease origami, Glen Whitney introducing me to the origami flat torus.
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Proceedings of the National Academy of Sciences , 109(19):7218–7223, 2012.[2] Dmitry Fuchs and Serge Tabachnikov. More on paperfolding.
The American Mathematical Monthly ,106(1):27–35, 1999.[3] Thierry Henocque and Philippe Marin. Method for 3D printing of highly complex geometries: Thefirst" flat torus" printed in 3D. 2013.[4] David A. Huffman. Curvature and creases: A primer on paper.
IEEE Transactions on computers ,(10):1010–1019, 1976.[5] Nicolaas H Kuiper. On C1-isometric imbeddings. In
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