The backdrop for Charlie Gunn's and Delle Maxwell's computer animation "Not Knot" is the hyperbolic space. The story gives the mathematical explanation of knots.

"Not Knot" tells the story of one way that mathematicians understand knots. In its complete form, it lasts over 15 minutes; the version that was submitted to Prix Ars Electronica is an excerpt featuring the highlights of the complete version.

The project was initiated to visualize some of the exciting results in three dimensional topology made by mathematicians working with the Geometry Center, particularly results of William Thurston in the classification of three-dimensional spaces. His deep geometric intuition was, we felt, well-suited to the medium of computer visualization.
We chose to feature hyperbolic space because it has great appeal to both mathematical and nonmathematical audiences. It allows people to experience the concept of curved, or non-Euclidean, space in a realistic way, a concept which is of central importance in many physical theories of natural science. It is an area which can now be visualized with computer graphics for the first time, and as such is a powerful argument for the role of computer graphics as a window into unseen worlds. It is particularly fitting that 1992 Is the 200th anniversary of the birth of Nikolai Lobachevski, the Russian mathematician and educator who co-discovered hyperbolic geometry in the early 19th century.

The movie begins by introducing the world of knots and showing simple properties and definitions. The movie then asks how to understand the complement of a knot. It begins with the simpler question of how to understand the complement of a line. Once this question has been answered, it turns to the more difficult question of understanding the complement of a more complicated knot, a set of three interlocked circles known as the Borromean rings. This quest leads the viewer out of everyday space into the strange and fascinating world of hyperbolic geometry, including a flight through a right-angled dodecahedral tessellation of hyperbolic space.29.02.1996The movie was created over the space of eighteen months from November, 1989 to May, 1991 by a collaborative group working at the Geometry Supercomputer Project at the University of Minnesota, now known simply as the Geometry Center. Group members brought together skills from mathematics, computer animation, art, design, and computer science.

The movie avoids technical language throughout, even while illustrating advanced mathematical concepts. Our aim was to demonstrate to the mathematical community that computer graphics is mature enough to communicate advanced concepts without compromising their depth. We also aimed to reach a non-mathematical audience with the message "Mathematics is exciting and visual".

Technical Background
15 min
SW: Softimage (Inc. Montreal), Pixar (Inc. Richmond),Wolfram Research, Champain (IL), Raysmade Craig Kolb

Hyperbolic Space in "Not Knot"
Dana S. Scott

What has been accomplished brilliantly by computer graphics and animation in this video might be described as "virtual unreality", in the sense that the viewer is given the impression of what it is like to fly through hyperbolic space within the lattice work of several kinds of non-Euclidean regular tilings of space. (Of course, it would be even more fun to fly an interactively simulated hyperbolic space ship in real time, but such a feat takes a large amount of computer power.) Even if we lived in such a non-Euclidean space, we would see such a small part of that space there would be hardly any perceivable difference from the Euclidean geometry of school books.

Thus, the effect of the graphics of "Not Knot" is to present hyperbolic geometry in truly cosmic proportions in order to show the important differences from familiar space. The imaginary geometry is made to seem very real and visually most disturbing, owing to continual changes in the apparent sizes of objects.

The well known graphic artist M. C. Escher drew many pictures of hyperbolic tilings of two-dimensional space, and it is safe to say he would have been very happy indeed to move into three dimensions using the graphic facilities that made this video possible. The principal objective of "Not Knot" is not really artistic, however, but rather to present in an attractive form some quite advanced mathematical ideas from topology and geometry. It is very difficult to make such mathematical concepts even partially understandable to general audiences, and a great amount of pedagogical insight has been brought to bear on this production. Surely one of the main aims of the creators is to make people want to learn more.

The story the video is meant to illustrate is how concepts of topology and geometry give us mathematical tools to demonstrate why different knots and links made up of closed curves are fundamentally different. We may ask, "what is 'topology'?" In short, topology is the study of connectedness and continuous passage between points, lines, surfaces and solids. "Geometry" adds to topology notions of measure and size: length, area, volume, angle, curvature. Topologically speaking there is no difference between hyperbolic and Euclidean space, since both kinds of space are topologically equivalent to the (finite) space inside a (Euclidean) solid sphere (billiard ball) or an ellipsoid (egg). But metrically speaking, all these spaces are different: Euclidean space is infinite, but the solid sphere is finite; Euclidean space is flat, but hyperbolic space is curved (in a suitable mathematical sense which gives it a uniform negative curvature of-1).

How do we get from knots to geometries? Knots exist in, say, ordinary Euclidean space, and the question of whether a knot can be untied or shown to be equivalent to another knot -without cutting or going "outside" the given space - has a precise topological definition. The only problem is that the deciding of equivalence of knots is not at all easy - neither practically nor theoretically.

The first major step to understanding the problem was to see that the space outside the knot was important. (This is the part of Euclidean space left after a worm eats out the knot. It is the pan that is not the knot.) The recognition that a solid space like this had to be considered took place some decades ago, but the precise proof that the topological type of the not knot space characterized knots up to tying-and-untying equivalence was only done in 1989! In the case of links, the external space does not characterize them, but this is a rather technical question, and the outside space is still quite important.

Where does geometry come in? So far we have only been talking topologically. In understanding spaces, it helps very much to represent them as geometric objects with metrical properties. The same space may have several representations. For example, the surface of a sphere and the surface of a cube art topologically equivalent; however, the very symmetrical spherical representation shows u: in visual form how very many useful continuous: transformations there are for such a space - such as rotations or mirror reflections. In the "No, Knot" video it is shown how to represent the spaces of certain links in a very nice geometric form (with very much symmetry) by using hyperbolic space rather than Euclidean space.

Why is hyperbolic space so interesting? Aside from giving a meaning to the word "geometry" that establishes the independence of Euclid's Parallel Postulate - a problem that gave mathematicians sleepless days for many centuries - hyperbolic space admits a vast number of kinds of tilings or regular coverings impossible in Euclidean space. For example, with regular tiles (polygons) we can tile the Euclidean plane only with regular triangles, squares and hexagons. The regular pentagon will not cover the Euclidean plane without overlaps. In hyperbolic space, however, we can cover a plane with regular pentagons, or with regular n-gons with any number of sides. This was a fact Escher knew and exploited. More generally, this is a fact that allows hyperbolic space to represent many kinds of symmetric patterns, when Euclidean space cannot do so without distortion.

In hyperbolic geometry, the sum of the three angles of a triangle is strictly less than 180°. In Euclidean geometry, the angle sum - as is well known - is exactly 180° (the parabolic case). In spherical geometry (with great-circle arcs as lines) the angle sum of a spherical triangle is strictly greater than 180° (the elliptic case). All these geometries allow for free mobility -running around in space without changing shape or size. But there is a rub in hyperbolic geometry: The angle sum of a triangle exactly determines its area, and so there are no similarities or magnifications. In hyperbolic space, a square with four right angles does not exist, and the best you can do is to have four equal angles, each less than 90°. How much less determines the size of the "square" and the lengths of its sides. The relationship between length and area or volume is different in the hyperbolic case from the Euclidean case as well. As they say on the video box: "A hyperbolic hemispherical swimming pool 25 meters in diameter contains 23 times the (ordinary) volume of the earth." In some sense "there is more room to fool around" in hyperbolic space, and thus it is more convenient for certain kinds of representations.

Is hyperbolic space unreal? Well, it is unreal in the sense that it is unfamiliar, but it is perfectly consistent as a geometric system - and it is very beautiful in its consistency and symmetry. Philosophically, Euclidean space is certainly unreal as well, in that it is a mathematical abstraction. It seems familiar, since in ordinary life and in Platonic / Newtonian cosmology we find it a simplifying and helpful assumption that space is Euclidean. The consistency of hyperbolic geometry can actually be demonstrated with a Euclidean representation; in fact, that is what is used in the computer graphics of this video.

In its simplest form, the hyperbolic plane can be represented as the interior of a circular disk, where the lines of the geometry are taken to be arcs of circles perpendicular to the boundary circle of the disk. In this representation, the Euclidean angles of intersecting circular arcs exactly represent the correct hyperbolic angles; but distances get very distorted (for the boundary circle must be thought of as being infinitely far away from the center point). And the "lines" look "curved" in the representation. But this is just a representation. The properties are correct, even if the representation looks funny.

This representation is also the one chosen by Escher for some of his drawings. In the video the creators had to go an important step beyond this representation to compute hyperbolic optics and perspective in order to represent the act of flying through space. The "inhabitants" of hyperbolic space find that things look different from what their Euclidean counterparts might , expect. This demonstration is a very interesting side-effect of the making of this animation.

What next? Well there are many other knots and links, many other patterns of tilings and lattices, and many other dimensions to explore. Now that we have a virtual camera onto these geometries, many other trips and slide shows should follow. And someone should take up Escher's "pen " and give us more of his kind of pictures. After all the mathematicians have already worked out the theory needed.

(Dana S. Scott, Professor for Computer Sciences, Philosophy and Mathematical Logic, Hillman University and Carnegie Mellon University)