The "Superellipse": a curve that lies between the ellipse and the rectangle.
SCIENTIFIC AMERICAN September 1965:
Civilized man is surrounded on all sides, indoors and out, by a subtle, seldom-noticed conflict between two ancient ways of shaping things: the orthogonal and the round. Cars on circular wheels, guided by hands on circular steering wheels, move along streets that intersect like the lines of a rectangular lattice. Buildings and houses are made up mostly of right angles, relieved occasionally by circular domes and windows. At rectangular or circular tables, with rectangular napkins on our laps, we eat from circular plates and drink from glasses with circular cross sections. We light cylindrical cigarettes with matches torn from rectangular packs, and we pay the rectangular bill with rectangular bank notes and circular coins.
Even our games combine the orthogonal and the round. Most outdoor sports are played with spherical balls on rectangular fields. Indoor games, from pool to checkers, are similar combinations of the round and the rectangular. Rectangular playing cards are held in a fan-like circular array. The very letters on this rectangular page are patchworks of right angles and circular arcs. Wherever one looks the scene swarms with squares and circles and their affinely stretched forms: rectangles and ellipses. (In a sense the ellipse is more common than the circle, because every circle appears elliptical when seen from an angle.) In the new Op paintings and textile designs, discussed here in July, squares, circles, rectangles and ellipses jangle against one another as violently as they do in daily life.
The Danish writer and inventor Piet Hein recently asked himself a fascinating question: What is the simplest and most pleasing closed curve that mediates fairly between these two clashing tendencies? Originally a scientist, Piet Hein (he is always spoken of by both names) is well known throughout Scandinavia for his 24 enormously popular volumes of gracefully aphoristic poems (which critics have likened to the epigrams of Martial) and for his writings on scientific and humanistic topics. To readers of this department he is best known as the inventor of the game Hex (July, 1957), of a new game similar to nim (February, 1958), of the Soma cube (September, 1958), of a braiding game (December, 1959) and of other remark-able recreations. He was a friend of the late Norbert Wiener, whose last book, God and Golem, Inc., is dedicated to him.
The question Piet Hein asked him-self had been suggested by a knotty city-planning problem that first arose in 1959 in Sweden. Many years earlier Stockholm had decided to raze and re-build a congested section of old houses and narrow streets in the heart of the city, and after World War II this enormous and costly program got under way [see "Stockholm: A Planned City," by Göran Sidenbladh, page 106]. Two broad new traffic arteries running north-south and east-west have been cut through the center of the city. At the intersection of these avenues a large rectangular space about 200 yards long is being laid out. At its center will be an oval basin with a fountain surrounded by a large oval pool containing several hundred smaller fountains. Daylight will filter through the pool's translucent bottom into an oval self-service restaurant, below street level, surrounded by oval rings of pillars and shops. Below that there will be two more oval floors for dining and dancing, cloakrooms and kitchen.
In planning the exact shape of this center the Swedish architects ran into unexpected snags. The ellipse had to be rejected because its pointed ends would interfere with smooth traffic flow around it; moreover, it did not fit harmoniously into the rectangular space. The city planners next tried a curve made up of eight circular arcs, but it had a patched-together look with ugly "jumps" of curvature in eight places. In addition, plans called for nesting different sizes of the oval shape, and the eight-arc curve refused to nest in a pleasing way.
At this stage the architectural team in charge of the project consulted Piet Hein. It was just the kind of problem that appealed to his combined mathematical and artistic imagination, his sense of humor and his knack of thinking creatively in unexpected directions. What kind of curve, less pointed than the ellipse, could he discover that would nest pleasingly and fit harmoniously into real values of x and y that satisfy the equation (in modern jargon, its "solution set") determine the points on the graph that lie on an ellipse with its center at the origin of the two coordinates. As n decreases from 2 to 1, the oval be-comes more pointed at its ends ("sub-ellipses," Piet Hein calls them). When n = 1, the figure is a parallelogram. When n is less than 1, the four sides are concave curves that become increasingly concave as n approaches 0. At n = 0 they degenerate into two crossed straight lines.
If n is allowed to increase above 2,
the rectangular open space at the heart of Stockholm?
To understand Piet Hein's novel answer we must first consider the ellipse, as he did, as a special case of a more general family of curves with the following formula in Cartesian coordinates,
an + bn 1 '
where a and b are unequal parameters (arbitrary constants) that represent the two semiaxes of the curve, and n is any positive real number. When n = 2, the
the oval develops flatter and flatter sides, becoming more and more like a rectangle; indeed, the rectangle is its limit as n approaches infinity. At what point is such a curve most pleasing to the eye? Piet Hein settled on n = 232'. With the help of a computer, 400 coordinate pairs were calculated to 15 decimal places, and larger, precise curves were drawn in many different sizes, all with the same height-width ratios (to conform with the proportions of the open space at the center of Stock-holm). The curves proved to be strangely satisfying, neither too rounded nor too orthogonal, a happy blend of elliptical and rectangular beauty. Moreover, such curves could be nested, as shown in the illustrations on page 224 and above, to give a strong feeling of harmony and parallelism between the con-centric ovals. Piet Hein calls all such curves with exponents above 2 "super-ellipses." Stockholm immediately accepted the 2132'-exponent superellipse as the basic motif of its new center. The huge, partly subterranean structure is now under construction. When the en-tire center is finally completed-perhaps in 1967-it is expected to be one of the great tourist attractions (certainly for mathematicians!) of Sweden.
Meanwhile Piet Hein's superellipse has been enthusiastically adopted by Bruno Mathsson, a well-known Swedish furniture designer. He first produced a variety of superelliptical desks, now in the offices of many Swedish executives, and has since followed with superelliptical tables, chairs and beds. (Who needs the corners?) Industries in Denmark, Sweden, Norway and Finland have turned to Piet Hein for solutions to various orthogonal-v. -circular problems, and in recent months he has been working on superelliptical furniture, dishes, coasters, lamps, silverware, textile patterns and so on. The tables, chairs and beds embody another Piet Hein invention: unusual self-clamping legs that can be removed and attached with great ease.
"The superellipse has the same convincing unity as the circle and ellipse, but it is less obvious and less banal," Piet Hein wrote recently in the leading Danish magazine devoted to applied arts and industrial design. (The magazine's white cover for that issue bore only the stark black line of a superellipse, captioned with the formula of the curve.)
"The superellipse is more than just a new fad," Piet Hein continued; "it is a relief from the straitjacket of the simpler curves of first and second powers, the straight line and the conic sections." Incidentally, one must not confuse the Piet Hein superellipse with the superficially similar potato-shaped curves one often sees, particularly on the face of television sets. These are seldom more than oval patchworks of different kinds of arc, and they lack any simple formula that gives aesthetic unity to the curve.
When the axes of an ellipse are equal, it is of course a circle. If in the circle's formula, x- + y2 = 1, the exponent 2 is replaced by a higher number, the graphed curve becomes what Piet Hein calls the "supercircle." At 2332 it is a genuine "squared circle" in the sense that it is artistically midway between the two extremes. The changing shapes of curves with the general formula x' + yn = 1, as n varies from 0 to infinity, are graphed below. If the graph could be stretched uniformly along one axis (one of the affine transformations), it would depict the family of curves of which the ellipse, sub-ellipses and super-ellipses are members.
In the same way one can raise the exponent in the corresponding Cartesian formulas for spheres and ellipsoids to obtain what Piet Hein calls "super-spheres" and "superellipsoids." If the exponent is 212, such solids can be regarded as spheres and ellipsoids that are halfway along the road to being cubes and bricks.
The true ellipsoid, with three unequal axes, has the formula (a/x) + (b/y) = c , where a, b and c are unequal parameters representing half the length of each axis. When the three parameters sire equal, the figure is a sphere. When only two are equal, the surface is called an "ellipsoid of rotation" or a spheroid. It is produced by rotating an ellipse on either of its axes. If the rotation is on the longer axis, the result is a pro-late spheroid-a kind of egg shape with circular cross sections perpendicular to the axis.
It turns out that a solid model of a prolate spheroid, with homogeneous density, will no more balance upright on either end than a chicken egg will, unless one applies to the egg a strata-gem usually credited to Columbus. Columbus returned to Spain in 1493 after having discovered America, thinking that the new land was India and that he had proved the earth to be round. At Barcelona a banquet was given in his honor. This is how Girolamo Benzoni, in his History of the New World (Venice, 1565), tells the story (I quote from an early English translation):
"Columbus, being at a party with many noble Spaniards ... one of them undertook to say: `Mr. Christopher, even if you had not found the Indies, we should not have been devoid of a man who would have attempted the same thing that you did, here in our own country of Spain, as it is full of great men clever in cosmography and literature.' Columbus said nothing in answer to these words, but having desired an egg to be brought to him, he placed it on the table saying: `Gentlemen, I will lay a wager with any of you, that you will not make this egg stand up as I will, naked and without anything at all.' They all tried, and no one succeeded in making it stand up. When the egg came round to the hands of Columbus, by beating it down on the table he fixed it, having thus crushed a little of one end; wherefore all remained confused, understanding what he would have said: That after the deed is done, every-body knows how to do it."
The story may be true, but a suspiciously similar story had been told 15 years earlier by Giorgio Vasari in his celebrated Lives o f the Most Eminent Painters, Sculptors and Architects (Florence, 1550). Young Filippo Brunelleschi, the Italian architect, had designed an unusually large and heavy dome for Santa Maria del Fiore, a cathedral in Florence. City officials had asked to see his model, but he refused, "proposing instead ... that whosoever could make an egg stand upright on a flat piece of marble should build the cupola, since thus each man's intellect would be discerned. Taking an egg, therefore, all those Masters sought to make it stand upright, but not one could find a way. Whereupon Filippo, being told to make it stand, took it graciously, and, giving one end of it a blow on the flat piece of marble, made it stand upright. The craftsmen protested that they could have done the same; but Filippo answered, laughing, that they could also have raised the cupola, if they had seen the model or the design. And so it was resolved that he should be commissioned to carry out this work."
The story has a topper. When the great dome was finally completed (many years later, but decades before Columbus' first voyage), it had the shape of half an egg, flattened at the end.
What does all this have to do with supereggs? Well, Piet Hein (my source, by the way, for the references on Columbus and Brunelleschi) discovered that a solid model of a 2'2-exponent superegg-indeed, a superegg of any exponent-if not too tall for its width, balances immediately on either end without any sort of skulduggery! In-deed, dozens of chubby wooden and silver supereggs are now standing politely and permanently on their ends all over Scandinavia.
Consider the wooden superegg shown on this page, which has an exponent of 2,5 and a height-width ratio of 4:3. It looks as if it should topple over, but it does not. This spooky stability of the superegg (on both ends) can be taken as symbolic of the superelliptical balance between the orthogonal and the round, which is in turn a pleasant symbol for the balanced mind of individuals such as Piet Hein who mediate so successfully between C. P. Snow's "two cultures." With Piet Hein's cooperation, the late just Lunning, president of Georg Jensen Inc. in New York City, had a balancing silver superegg made with an exponent of 2,5 and a height-width ratio of 6:5, 15 centimeters high and 12 centimeters wide. (A similar egg is shown in the illustration on page 234.) He offered it as an animal prize at Herlufholm, Denmark's Eton, as an award to the student who has made the most out-standing contribution uniting widely different fields. Piet Hein, who has lectured and written extensively about balancing the scientific and humanistic outlooks, is himself planning an international prize, to be symbolized by a golden superegg, for achievements that promote such balance.
These 23-exponent superellipsoids of rotation are not the only ones that form solid eggs that balance upright on either end; similar balancing eggs can be constructed by rotating superellipses with any exponent above 2. The higher the exponent, the taller and thinner such a superegg can be before it reaches the critical height-width ratio at which it topples over. At the limit, when the exponent is infinity, an "egg" only one centimeter wide can be made as tall as the Empire State Building and still, in theory, remain stable. Such an "egg," generated by the rotation of a rectangle, is simply a right-circular cylinder with a perfectly flat circular base. Clearly there is no limit to the height-width ratio at which such a cylinder is stable on either end.
Without getting into the complexities of calculus, can the reader prove that whenever the exponent in a superegg's formula is greater than 2, it is possible to construct stable eggs that are taller than they are wide? Piet Hein's simple proof will be given next month.