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Turning a Surface Inside Out

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Turning a Surface Inside Out TURNING A SURFACE INSIDE OUT N orn1ally a sphere can be turned inside out only if it has been torn. In differential topology one assumes that the surface can be pushed through itself, but then the problem is to avoid forming a "crease" by Anthony PhiJlips he great mathematician David knurl-the remaining portion of the out­ face is then stretched, pinched and THilbert once said that a mathe­ side-vanishes. In the process, unfortu­ twisted through several intermediate matical theory should not be con­ nately, the knurl forms a tight loop that stages too complicated to depict in their sidered perfect until it could be ex­ must be pulled through itself [see up­ entirety. We can follow the changes plained clearly to the first man one met per illustration on opposite page]. This between successive stages by watching in the street. Hilbert's successors have results in a "crease" that is displeasing what happens to ribbon-like cross sec­ generally despaired of living up to this to differential topologists, whose dis­ tions of the surface as it is turned inside standard. As mathematics becomes more cipline is limited to smooth surfaces. out. It is possible to understand the specialized it is difficult for a mathe­ The problem for the differential top­ entire deformation by interpolating the matician to describe, even to his col­ ologist is how to turn the sphere inside missing parts of the surface at each leagues, the nature of the problems he out without introducing a crease in get­ stage and checking that the changes in studies. From time to time, however, ting rid of the knurl. Here too intuition the ribbons depicting various sections research on an advanced and inacces­ indicates that the problem cannot be fit together coherently. sible mathematical topic leads to a dis­ solved, and when Smale first announced covery that is intuitively attractive and he could prove that a solution existed, Turning a sphere inside out is only can be explained without oversimplifica­ even his thesis adviser at the Univer­ one example of the deformations tion. A striking example is Stephen sity of Michigan warned him that there shown to be possible by Smale's theo­ Smale's theorem concerning regular was an "obvious counterexample" to rem. To explain the full implications of maps of the sphere, published in 1959. his claim. Intuition was wrong; no fault the theorem we must introduce mathe­ The field in which Smale was then could be found with the logic of Smale's matical definitions of the objects and working-differential topology, which proof. In fact, it was theoretIcally pos­ deformations with which we are con­ combines concepts from topology and sible to follow the proof step by step cerned. Let us begin with some precise calculus-is one of the more abstract and to discover an explicit description descriptions of curves. The circle is domains of modern mathematics. Never­ of the deformation that turns the sphere defined to be the set of points in a two­ theless, a visualization devised by the inside out. The argument was so com­ dimensional plane (located by an x axis late Arnold Shapiro of Brandeis Uni­ plicated, however, that the actual task and a y axis) at distance 1 from the versity enables us to depict a startling seemed hopeless. For some time after origin (the intersection of the two axes). consequence of Smale's theorem: It is Smale's discovery it was known that the Having established the radius as 1, we possible, from the topologist's point of sphere could be turned inside out with­ can describe any point P on the circle view, to turn a surface such as a sphere out a crease, but no one had the slight­ by the angle e between the x axis and a inside out. est idea how to do it. line from the origin to P [see upper il­ How is this accomplished? It is intui­ This problem, concerning such a sim­ lustration on page 114]. In other words, tively clear that unless a sphere is some­ ple object as the sphere, was extremely any point on the circle can be described how torn it must remain right-side out, mystelOious and challenged the minds of by a number from 0 to 360 giving the no matter how one is allowed to deform many mathematicians. As far as I know number of degrees in the corresponding and displace it. If we are mentally the only ones who eventually worked it angle e. allowed to move the surface through out were Nicolaas Kuiper, who is now A curve in the plane will be defined itself, however, so that two points on it at the University of Amsterdam, and as a map from the circle into the plane. can occupy the same point in space (this Shapiro. Shapiro's idea of the deforma­ We shall consider several examples of is permissible in differential topology), tion provides the basis for the illustra­ such maps; each of them is a rule as­ then a solution suggests itself. It is the tion that begins at the bottom of the signing to each point of the circle (or, deformation in which two regions of the opposite page and ends on page 117. equivalently, to each number e be­ sphere are pushed toward the center The deformation of the gray sphere be­ tween 0 and 360) a point in the plane. from opposite sides until they pass gins with the pushing of two regions on If c is such a map, then the point it through each other. The original inner opposite sides of the sphere toward the assigns to e is denoted c (e) and is " surface begins to protrude in two places, center and through each other so that called "the image of e under c. Natu­ which are then pulled apart until the the colored interior protrudes. The sur- rally we require that c (0) equal c (360), 112 © 1966 SCIENTIFIC AMERICAN, INC 2 3 INADMISSIBLE PROCEDURE for turning a sphere inside out en· is pulled through itself, a "crease" is introduced in the surface; this tails pushing regions on opposite sides toward the center (2) and violates a law of differential topology, a discipline of mathematics through each other. The original interior (color) hegins to pro· that is concerned only with smooth surfaces. In this discipline trude on two sides (3); these two sides are pulled out to form a moving a surface through itself is permissihle. The ribbons at bot· sphere (4 and 5). When the looped portion of the original sudace tom depict a section of the surface during stages of deformation. since 0 and 360 describe the same point helpful to think of the path of a mov­ sin (). (If () is taken as an acute angle on the circle. ing particle. This could be made into of a right triangle, then sin (), the sine Why should one give such abstract a definition more easily grasped by the of (), is the ratio of the length of the and complicated meanings to such intuition than "a map from the circle opposite leg to the length of the simple terms as "curve"? There are two into the plane," but the more abstract hypotenuse; cos (), the cosine of (), is important reasons. First, precise defini­ definition will enable us to generalize the ratio of the length of the adjacent tions are essential for a sound mathe­ our study of curves to that of two­ leg to the length of the hypotenuse.) matical theory. Second, a good defi­ dimensional surfaces, our main interest. This description can then be abbrevi­ nition will suggest by analogy how a A simple example of a curve is pro­ ated to c (()) = (2 cos (), sin ()). theory can be extended to encompass vided by the map c that assigns to each The curve defined by g (()) = (cos (), new material. For example, to get a () the point in the plane with an x co­ sin ()) is even simpler. If the point P picture of a curve in the plane it is ordinate of 2 cos () and a y coordinate of on the circle corresponds to the angle (), D E F B HOW TO TURN A SPHERE INSIDE OUT by steps that conform One part of the original interior is then distended (C) to give to the laws of differential topology is depicted in this illustration the surface resembling a saddle on two legs (D). The two legs and those at bottom of the next four pages. The deformation begins are then twisted counterclockwise to give surface E. This surface with the pushing of opposite sides toward the center and through is shown again (F) with ribbons depicting it in cross section on each other so that the interior (color) protrudes in two regions (B). different levels. Thin black line indicates missing parts of surface. 113 © 1966 SCIENTIFIC AMERICAN, INC y y y -t--------------------+-------------------�-x -t--------�f_--------+_-x [OJ g(8) = (cos 8, sin 8) C(8) = (2 cos 8, sin 8) h(8) = (sin 8, sin 28) CURVES IN THE PLANE are defined and exemplified. The circle the x axis and the line from the origin to P. A closed curve in the is defined as the set of points in a two·dimensional plane at dis· plane can be given by a map that assigns to each number e a point tance I from the origin (the intersection of the x axis and the y in the plane. The map c that assigns to each e the point whose x axis).
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