Wave, Boy’s Surface, and Machine

Year: 2022 Authors: Annie Wei; Yongheng Zhang

Core claim

Adding period-3 corrugation to a base sphere map yields an immersed Boy’s surface and motivates a light-driven drawing machine.

Topics

Boy’s surface, Thurston corrugation, sphere eversions, immersed surfaces

Domains

differential topology, geometry, surface immersions, real projective plane, kinetic sculpture, machine drawing, visualization, light-based installation

Methods

analytical formula, corrugation technique, period-3 waves, orthonormal frame field

Media

electronic lights, space drawing machine, surface diagrams, parametric curves

Paper text

The text below is the locally extracted OCR/Markdown version of the paper. Raw PDF files remain local and are not published here.

Bridges 2022 Conference Proceedings

Wave, Boy’s Surface, and Machine

Annie Wei ${}^1$ and Yongheng Zhang ${}^2$

Department of Mathematics and Statistics, Amherst College, MA, USA ${}^1$ arwei25@amherst.edu, ${}^2$ yzhang@amherst.edu

Abstract

We describe an analytical method of constructing Boy’s surface which uses Thurston’s corrugation technique to add period-3 waves to an image of the 2-sphere. This construction allows us to build a simple machine which can draw Boy’s surface in space.

Introduction

A hundred and twenty-two years ago, David Hilbert suspected that it is impossible to immerse the real projective plane $\mathbb{R}{P}^2$ in ${\mathbb{R}}^3$ . He asked his doctoral student Werner Boy to prove it. Instead, Boy discovered the opposite, finding an immersed image of $\mathbb{R}{P}^2$ , later called Boy’s surface. Boy’s description was qualitative, relying mostly on pictures. Afterwards, analytical formulas were found, such as those by Morin, Petit-Souriau, Hughes, Bryant-Kusner, and Apéry, summarized in the book [1]. In this paper, we present another analytical formula, which inspired the construction of a simple machine that draws Boy’s surface in space.

To understand the abstract surface $\mathbb{R}{P}^2$ , imagine the planet Earth, modeled by the unit sphere $S^2$ in ${\mathbb{R}}^3$ , with the north pole $N$ at $(0,0,1)$ and the south pole $S$ at $(0,0,-1)$ . On $\mathbb{R}{P}^2$ , these two poles are considered the same point, and in general, $\mathbb{R}{P}^2$ identifies any two antipodal points on $S^2$ . It is not immediately clear that $\mathbb{R}{P}^2$ , an abstract geometric space, can be immersed in ${\mathbb{R}}^3$ , as its representation would possess qualities which are difficult to visualize. For example, unlike $S^2$ , $\mathbb{R}{P}^2$ contains a Möbius strip, so it must intersect itself as a closed surface in ${\mathbb{R}}^3$ [8]. Additionally, the surface must contain a triple point [2].

Figure 1: Boy’s surface is an image of $S^2$ in ${\mathbb{R}}^3$ ; The formula $B=b+W_3$ .

The Boy’s surface constructed in this paper is an immersed image of a smooth map $B:S^2\to {\mathbb{R}}^3$ with each pair of antipodal points mapped to the same location. $B$ is given by the formula

where $b:S^2\to {\mathbb{R}}^3$ is a base map collapsing the equator to a single point, and is thus not an immersion. To remove this singularity without creating new singularities, we add $W_3:S^2\to {\mathbb{R}}^3$ , a period-3 wavy surface which wraps around the image of $b$ via an orthonormal frame field, and whose amplitude grows from zero at the poles to a maximum on the equator. This method of viewing a surface as the composition of a simpler

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base surface and a wavy surface (Figure 1) is due to William Thurston. It was used in his method of everting the sphere, as illustrated in the movie Outside In [6].

In this paper, we first review the analytical theory of Thurston’s corrugation method [7] using curves in the plane, as an analogy for understanding the subsequent 3D construction of the Boy’s surface. Then we describe a machine inspired by Formula (1), which is used to paint Boy’s surface in the dark using electronic lights.

Wave

Consider the parametrized curve $f(t)=⟨y(t),z(t)⟩$ on the $yz$ plane:

(a) The components $y(t)$ and $z(t)$ are $C^1$ .

(b) But $f^{\prime }(t)=⟨0,0⟩$ at the four vertices. Figure 2: The rectangular curve $f(t)=⟨y(t),z(t)⟩$ is $C^1$ but not regular.

The curve $f$ is $C^1$ , as both derivatives $y^{\prime }(t)$ and $z^{\prime }(t)$ exist and are continuous (Figure 2a). However, $f$ is not regular as $f^{\prime }$ vanishes at four points. Its image is a rectangle, with four sharp corners (singularities) at points where $f^{\prime }(t)=⟨0,0⟩$ (Figure 2b). We will remove the singularities of $f$ by adding waves, which we construct using the following periodic curve $8:\mathbb{R}\to {\mathbb{R}}^2$ :

As $t$ increases from 0 to $2\pi$ , 8 begins at the origin, moves northwest, then traces an “8” on the $yz$ plane (Figure 3a). The derivative $8^{\prime }$ lies in an annular sector missing the quarter symmetric about the negative $y$ -axis (Figure 3b). This feature is important for maintaining regularity when the amplitude of waves become variable and small, especially near the poles of the base map $b$ in Formula (1) as we’ll see in the next section.

Now consider $a8(nt)$ , which is 8 with adjusted amplitude $a$ and frequency $n$ of oscillation. Adding to $f(t)$ gives:

shown in Figure 4a. To see that $F_1(t)$ is regular, note $f^{\prime }(t)$ is continuous over the compact domain $[0,2\pi ]$ , so $f^{\prime }(t)$ is bounded, and we may choose $a$ and $n$ such that an large causes $an{8}^{\prime }(nt)$ to lie outside a sufficiently big circle containing $f^{\prime }(t)$ in its interior. This ensures |an\mathbf{8}'(nt)| > |f^{\prime}(t)| everywhere, which implies that $F_1^{\prime }(t)$ is never zero. For $F_1$ to keep semblance with $f$ , the amplitude $a$ of the waves cannot be too large, but this is compensated with big $n$ . Note that adding 8 may change a curve from an embedding to an immersion, as is the case for $F_1$ .

Wave, Boy’s Surface, and Machine

(a) Image of 8.

(b) Image of 8’.

(c) $G(t)=g(t)+R(t)8(3t)$ . Figure 3: The model figure-8 curve and its derivative used in this paper; Planar “trefoil”.

When constructing the 3D Boy’s surface, pointwise variation of amplitude and pointwise rotation of waves are also necessary. In the 2D case, the former can be done using a $C^1$ function $A(t)$ :

which is illustrated in Figure 4b with $A(t)=1+1/2\cos 4t$ . The latter can be done using a $C^1$ map $R:\mathbb{R}\to SO(2)$ , where $SO(2)$ is the space of $2\times 2$ orthonormal matrices with determinant 1:

The result of pointwise rotation is shown in Figure 4c with $R(t)=\left[\begin{array}{cc}\cos t& amp;-\sin t\\ \sin t& amp;\cos t\end{array}\right]$ .

(a) $F_1(t)=f(t)+a8(nt)$ Figure 4: Corrugated rectangles where $a=0.1$ and $n=50$ .

(b) $F_2(t)=f(t)+aA(t)8(nt)$ .

(c) $F_3(t)=f(t)+aA(t)R(t)8(nt)$ .

The formula for Boy’s surface to be discussed in the next section is a 3D version of $F_3(t)$ using corrugation with much lower frequency ( $n=3$ ). The corrugation, whose waves expand and rotate in 3D space, is used to remove a singularity that is the image of the equator. Its effect can be understood by the 2D example in Figure 3c where $g(t):=⟨0,0⟩$ maps a circle to a single point, and is thus not regular. The addition of $R(t)8(3t)$ introduces three uniformly rotated “8”s to $g$ so that $G$ is regular, and whose corrugation is contained to the plane. In 3D, the extra dimension provides space where the image of the equator can be folded into a twice traversed trefoil.

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Boy’s Surface

Consider the sphere $S^2$ as our planet Earth and let $\varphi$ be the polar angle, i.e., the angle of latitude, which is $\pi /2$ , 0 and $-\pi /2$ , at the north pole, the equator and the south pole, respectively. Let $\theta$ be the azimuthal angle, i.e., the angle of longitude, which is 0 for the prime meridian (the half circle from pole to pole intersecting the positive $x$ -axis), and is the same $\theta$ as in spherical coordinates. The definition of $\varphi$ differs from spherical coordinates, as rather than measured from the positive $z$ -axis downward, it is measured from the $xy$ -plane upward as shown in Figure 5a.

(a) Definition of $\theta$ and $\varphi$ .

(b) Image of $b$ Figure 5: The sphere $S^2$ , the image of the base map $b$ , and the Boy’s surface in this paper.

(c) Image of $B$

Define the base map $b:S^2\to {\mathbb{R}}^3$ as:

where $r(\theta )$ is the rotation matrix about the $z$ -axis through angle $\theta$ , i.e. $r(\theta )=\left[\begin{array}{ccc}\cos \theta & amp;-\sin \theta & amp;0\\ \sin \theta & amp;\cos \theta & amp;0\\ 0& amp;0& amp;1\end{array}\right]$ , and $l(\varphi )=⟨0,0,-\frac{1}{2}⟩+\frac{1}{2}⟨-\sin 2\varphi ,0,\cos 2\varphi ⟩$ , with “ $⟨$ ” and “ $⟩$ ” enclosing column vectors. The image of $b$ is shown in Figure 5b with $\theta$ and $\varphi$ values.

One can check that $b(\theta +\pi ,-\varphi )=b(\theta ,\varphi )$ , so $b$ maps antipodal points on $S^2$ to the same point in ${\mathbb{R}}^3$ . Thus, the image of $b$ is a representation of $\mathbb{R}{P}^2$ in ${\mathbb{R}}^3$ . Note $b$ is regular every where except on the equator $\varphi =0$ , which is mapped to a single point by $b$ . To remove this singularity, we add waves to the base map $b$ .

Let $8:\mathbb{R}\to {\mathbb{R}}^3$ be defined by $8(t)=⟨0,-\sin 2t,2\sin t⟩$ . This is the $8$ in the previous section as a spatial curve, and we add it to $b$ after adjusting its orientation, amplitude, and period.

(a) Orthonormal frames along $l(\varphi )$ .

(b) We use $\lambda (\varphi )$ to slow down the turning of frames near the poles.

(c) The amplitude $aA(\varphi )$ seen on $l(\varphi )$ . Figure 6: The frame $R(\varphi )$ , the saturation function $\lambda (\varphi )$ and the amplitude adjusting function $A(\varphi )$ .

(d) We use $A(\varphi )$ to adjust the amplitude of waves, which is 0 at the poles.

An orthonormal frame field $R(\varphi )=\left[T(\varphi ),U(\varphi ),V(\varphi )\right]$ , pictured in Figure 6a, rotates 8. The columns of $R(\varphi )$ are defined as follows.

Wave, Boy’s Surface, and Machine

First, $T(\varphi ):=l^{\prime }(\varphi )$ or the unit tangent vector along the image of the prime meridian:

Second, $U(\varphi )$ is a vector turning about $T(\varphi )$ , which we define using $N(\varphi )$ , a unit vector perpendicular to $T(\varphi )$ that points toward $(0,0,-1/2)$ , the center of the image of $b$ :

Then $U(\varphi )$ is a vector turning about $T(\varphi )$ on the plane spanned by $⟨0,1,0⟩$ and $N(\varphi )$ . As $\varphi$ goes from $-\pi /2$ to $\pi /2$ , $U(\varphi )$ completes a half twist from $⟨0,1,0⟩$ to $⟨0,-1,0⟩$ through angle $\pi$ . We also use $\lambda (\varphi )=\frac{\pi }{2}\sin (\varphi )$ to slow down the turning of $U(\varphi )$ near the poles (Figure 6b). Altogether, $U(\varphi )=-\sin \lambda (\varphi )⟨0,1,0⟩+\cos \lambda (\varphi )N(\varphi )$ , or:

Lastly, $V(\varphi ):=T(\varphi )\times U(\varphi )$ completes the frame:

We can think of $R(\varphi )$ as a miniature coordinate system, in which 8 is traced, oriented differently from point to point on the image of $l$ . $R(\varphi )$ is an element of $SO(3)$ , the space of orthonormal matrices with determinant 1.

At the moment, there is an ambiguity of $R(\varphi )$ at the poles after applying $r(\theta )$ , as $r(\theta )R(\pm \frac{\pi }{2})$ is different for different $\theta$ . To remedy this, we annihilate 8 at the poles using the $C^1$ function $A(\varphi )=\frac{1}{2}+\frac{1}{2}\cos 2\varphi$ . In Figure 6d we see $A(\varphi )$ grows from zero at the poles of $S^2$ (with derivative zero) to a maximum value on the equator. Figure 6c shows the resulting change in the image. The maximum amplitude $a$ is set to be 0.25 in Figures 1, 5c, 7b, and the second row of 8, though it can vary through a small range of values.

Putting these pieces together, let $W_n(\theta ,\varphi )=aA(\varphi )r(\theta )R(\varphi )8(n\theta )$ , with $n=3$ to create the three fold symmetry in Boy’s surface. Then

can be written as

Both $b$ and $W_3$ map antipodal points to the same point, so $B$ does as well, thus the image of $B$ is a representation of $\mathbb{R}{P}^2$ in ${\mathbb{R}}^3$ .

(a) The Bryant-Kusner Boy’s surface.

(b) The Boy’s surface constructed in this paper. Figure 7: Top, bottom and side views of Boy’s surfaces

In Figure 7 we see that the Bryant-Kusner Boy’s surface [5] and our constructed surface look similar. In fact, horizontal cross sections (on $z=k$ ) of the image of $B$ are ambient smooth isotopic to the cross sections of the Bryant-Kusner Boy’s surface (Figure 8 on the next page, where double points are labelled in red). The image of $B$ near the poles is almost a spherical cap, on which analysis with the key fact $A^{\prime }(\pm \pi /2)=0$ shows that $\frac{\partial B}{\partial x}$ and $\frac{\partial B}{\partial y}$ are never zero or colinear. Away from the poles, computer computation verifies that for any $a$ between 0.1 and 0.28, $\frac{\partial B}{\partial \theta }$ and $\frac{\partial B}{\partial \varphi }$ are nowhere zero, and the angle between them is never 0 or $\pi$ . Thus, the image of $B$ is indeed a Boy’s surface.

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Figure 8: Cross sections of Boy’s surfaces. Top: Bryant-Kusner; Bottom: in this paper.

Machine

Formula (1) implies that Boy’s surface can be drawn by two rotating vectors, added tip to tail. Inspired by this idea, a machine was constructed in which a rigid arm traces the spherical image of $b$ while an attached, rotating screen draws $W_3$ . One can check that

where $\infty (t)=⟨0,-2\sin t,\sin 2t⟩$ , and ${\mathrm{Rot}}^z(\theta )$ , ${\mathrm{Rot}}^{-y}(2\varphi )$ , and ${\mathrm{Rot}}^{-x}(\lambda (\varphi ))$ describe the counter clockwise rotation about the $z$ -axis, and the clockwise rotations about the $y$ and $x$ -axis respectively. Effectively, $l(\varphi )$ is $⟨0,0,1/2⟩$ rotated by ${\mathrm{Rot}}^{-y}(2\varphi )$ with tail shifted to $⟨0,0,-1/2⟩$ , and the action of $R(\varphi )$ has been decomposed into the actions of ${\mathrm{Rot}}^{-y}(2\varphi )$ and ${\mathrm{Rot}}^{-x}(\lambda (\varphi ))$ .

(a)

(b) https://www.youtube.com/watch?v=txa7jdZYw7M Figure 9: Conceptual mechanism and practical machine for drawing Boy’s surface.

Formula (2) describes the forward kinematics of an idealized mechanism pictured in Figure 9a. As the robotic “arm” moves about the ball-joint hinged at $(0,0,-\frac{1}{2})$ , the screen turns about the axis perpendicular to the arm and lying on the vertical plane, and a single glowing point traces out 8. Imagine the point as a toy car dashing around this figure-8 track. This was the analogy used in [7].

In the practical design of the machine (Figure 9b), three motors were used to produce the necessary motion: 1) the base stepper generates uniform speed rotation ${\mathrm{Rot}}^z(\theta )$ ; 2) the top micro servo controls ${\mathrm{Rot}}^{-y}(2\varphi )$ ; 3) the bottom micro servo holding the screen controls ${\mathrm{Rot}}^{-x}(\lambda (\varphi ))$ .

In Figure 9b, electronic and mechanical parts from Adafruit are held on a 3D-printed white frame. A single Arduino Uno board was sufficient for control and communication, and images were captured using long exposure photography. We programmed the machine to draw the image of $B$ , one parallel at a time. At different $\varphi$ values, the orientation of the screen and the amplitude of the figure-8 differ.

Wave, Boy’s Surface, and Machine

The resulting images for $a\approx 0.13$ are shown in Figure 10 and Figure 11, in which we see that as $\varphi$ gets closer to 0, the image circle eventually folds into a twice traversed trefoil. This trefoil is different from the trefoil curve of self-intersection of the image of $b$ , though they share the triple point at the origin.

Figure 10: Top views of images of the parallel $\varphi =\frac{i\pi }{40}$ .

Figure 11: Side views of images of the parallel $\varphi =\frac{i\pi }{40}$ .

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Notes

The method we used in this paper can be extended to construct a two-parameter family of immersed $\mathbb{R}{P}^2$ in ${\mathbb{R}}^3$. One parameter $n$ ($\ge 3$) is the period of the waves and the other parameter $m$ is the number of half twists of $R(\varphi )$ (following the right hand rule). Both $n$ and $m$ must be odd (and $m$ can be negative) so that antipodal points are identified. The Boy’s surface in this paper consists of one half twist ($m=1$) and period three waves ($n=3$), thus among these immersions of $\mathbb{R}{P}^2$, Boy’s surface is the simplest. Furthermore, for any $m$, $\epsilon -\delta$ proofs can be used to show that there exists an $N$ large enough such that for any $n\ge N$, the surface${}^2$ for $m$ and $n$ is an immersed $\mathbb{R}{P}^2$.

These immersions can be used as halfway models for sphere eversions [9]. This was the original use of Thurston’s corrugation technique. For any given $m$ and sufficiently large $n$, an analytical formula exists (similar to the one in this paper). Starting with this formula, $S^2$ can be smoothly deformed into the given immersion of $\mathbb{R}{P}^2$: the principle meridian regularly deforms to the image of $l$, such that the north pole remains on the $z$-axis while the south pole is stationary; meanwhile, the tangent frame of $S^2$ is twisted into $r(\theta )R(\varphi )$ about the tangent vectors of meridians. Since singularities only occur with respect to $\theta$, a single layer of corrugation is sufficient (compare with [3]${}^3$). Once the immersion of $\mathbb{R}{P}^2$ is reached, we play this movie backward with antipodal points traveling on swapped paths. Thus every point on $S^2$ trades positions with its antipodal counterpart, and the sphere has been turned inside out. Note that the sphere eversions obtained from different $m$ values can not be deformed into each other. They are like an infinite sequence of adaptations of a movie, with pairwise distinct plot twists.

Acknowledgements

We would like to thank the referees and the program chair David Reimann for helping us improve the presentation of this paper.

References

[1] F. Apéry. Models of the Real Projective Plane. Braunschweig; Wiesbaden: Vieweg, 1987. [2] T. F. Banchoff. “Triple points and surgery of immersed surfaces.” Proc. Amer. Math. Soc., Vol. 46, pp 407-413, 1974. [3] V. Borrelli, S. Jabrane, F. Lazarus, and B. Thibert. “Flat tori in three-dimensional space and convex integration.” Proc. Nat. Acad. Sci. USA, Vol 109, pp 7218-7223, 2012. https://www.pnas.org/content/109/19/7218 [4] W. Boy. “Über die Curvatura integra und die Topologie geschlossener Flächen.” Math. Ann., Vol. 57, pp 151-184, 1903. [5] H. Karcher and U. Pinkall. “Die Boysche Fläche in Oberwolfach.” Mitteilungen der DMV, Issue 1, pp 45-47, 1997. https://www.mfo.de/about-the-institute/history/boy-surface/karcher_pinkallboy_surface.pdf [6] S. Levy, D. Maxwell, T. Munzner, et al. https://youtu.be/OI-To1eUtuU. The Geometry Center, 1994. [7] S. Levy and W. Thurston. Making Waves – A Guide to the Ideas behind Outside In. The Geometry Center, University of Minnesota, 1995. A K Peters/CRC Press, 2009. [8] H. Samelson. “Shorter Note: Orientibility of Hypersurfaces in ${\mathbb{R}}^n$.” Proc. Amer. Math. Soc., Vol. 22, pp 301-302, 1969. [9] J. M. Sullivan. http://torus.math.uiuc.edu/jms/Papers/isama/color/opt2.htm. 1999.

${}^2$We encourage the reader to plot the cross sections of these surfaces. You might be surprised by what you find. ${}^3$In contrast, infinitely many layers of corrugation have been used to achieve $C^1$ isometric embeddings.