A botanist analyzes a flower bud by embedding it in a plastic cube, then slicing the cube into thin sheets, which she mounts in glass slides. Examining the stack of slides in sequence reveals the interior geometry of the bud. As the botanist goes from the bottom to the top of the stack, she occasionally finds a "critical" slide, where the slice changes form. To get a good idea of the total shape of the bud, she has only to set aside those crucial slides and some representative intermediate slices on either side. Given such a set of slides, it would be possible to re-create a physical model of the bud or to re-create the image in one's mind.

Thanks to recent advances in technology, we no longer have to cut the actual object in order to get a sequence of images of the slices at different heights and in different directions. The first breakthrough was the CAT scan. Computer axial tomography provides X-ray pictures of slices perpendicular to a patient's spine. More recent techniques produce magnetic resonance imaging of a head's interior structure by showing slices not just in the axial direction, perpendicular to the spine, but also from ear to ear, the "sagittal" slices, or from the tip of the nose to the back of the head, the "coronal" slices (see images on the next page). These images show both the bone and the tissue, different shades and textures indicating the densities of various parts. We can mount the images on plastic sheets, place them in order with the proper separation, and interpolate physically or mentally to obtain a three-dimensional view of the original shape.

In all of these cases, we gain insight into a three-dimensional structure by looking at sequences of two-dimensional slices. If we really understand a solid object, we can predict the slices we will get if we cut it by a series of parallel planes.

A different sort of three-dimensional slide sequence arises when we stack two-dimensional images taken over a period of time. A young amoeba goes about the course of its life unaware that, poised above the petri dish that is its universe, a camera is making a record of its actions, taking a picture every few moments as the amoeba explores its neighborhood and ingests the food it finds. It grows, reaches maturity, and then a day after it first appeared, splits off into two new beings, each event in its life captured by the camera. A technician develops the film onto thin glass squares and stacks them in a tray to form a long prism. Inside that glass prism, we can see a three-dimensional wormlike shape that records the entire life history of the amoeba. To study any event in that history, we have only to select the appropriate slide to get a slice of the amoeba's life.

Edwin Abbott Abbott used slicing to describe communication between
different dimensions in *Flatland*. The great climactic event in that
story occurs when A Square is visited by a creature from a higher dimension, in this case from our third dimension. That event totally and irrevocably forces him to revise his understanding of the nature of reality. Imagine A Square as an amoeba-like creature floating on the surface of a still pond, unaware of the air above or the water below, conscious only of the superficial reality, the surface of the water. About to intrude on this two-dimensional universe is a sphere from the third dimension, a beach ball. The water parts as the sphere passes through the surface. To A Square, who can see only the part of the sphere intersecting his plane, the visitation is very mysterious. At each stage he sees only the edge of a plane figure, and he can walk around the figure to observe that it is a perfectly round circle. In *Flatland*, the circles have all the power, sacred and secular. They are the high priests and philosopher kings. Seeing the passage of the sphere through Flatland, A Square could describe it in only one way. He would say that he had just experienced the accelerated lifetime of a priest! First he would see a priestly zygote, which grows to a circular priestly embryo. A priestly infant is born and grows through minor orders into ordination and full monsignority, only to diminish in size as it reaches old age, until finally it shrinks down to a point and disappears. A Square experiences the successive slices of the sphere as a two-dimensional creature growing and changing in time. In Flatland, it takes quite an effort for A Sphere, himself a bit of a mathematician, to convince A Square that A Sphere is not a two-dimensional creature growing and changing in time, but a being that extends spatially beyond the two dimensions of Flatland into a third dimension unknown to the Flatlanders. A Square experiences that third dimension as time, but that does not mean that "the third dimension is time."

And what challenge does that give to us who are privileged to live in Spaceland? Here we are in our three-dimensional "still pond," ready to believe that this is all there is. What would happen if we were visited by a sphere from the fourth dimension? The analogy is clear. First we would see a point expanding in every direction we can see to form a small sphere, which grows until it reaches its full distention, then shrinks back down to a point and disappears. That same sequence of growth and decline can be simulated precisely by the gradual inflation and deflation of a balloon. Without more information, it would be impossible for us to tell whether we were seeing an ordinary sphere growing and changing in time or the successive three-dimensional slices of a "hypersphere" from the fourth dimension.

Modern readers are often perplexed by the significance of *time* as a fourth dimension. That is just what it is, *a* fourth dimension, not *the* fourth dimension. Abbott wrote *Flatland* twenty years before anyone had thought of relativity theory, where time is treated as a fourth dimension, so this was not a point of confusion for him, as it is for us who have been brought up in the twentieth century. Certainly people in the nineteenth century were aware that time often appeared in equations and that it could be represented spatially on graphs. People understood and used the idea of time as a fourth coordinate in something as prosaic as making an appointment in a place like New York City. "I'll meet you at the corner of Seventh Avenue and Fourth Street on the fifth floor" might be entered in the calendar as (7, 4, 5), but the appointment would not be complete without that fourth time coordinate, "at ten o'clock," which could be recorded as (7, 4, 5, 10). This four-dimensional system of "three-space, one-time" is immensely useful in modern physics, not just as a bookkeeping device but as a rich mathematical structure. The mathematics is not the same as that of ordinary space though; the time dimension really acts differently. What Abbott is presenting is the challenge of imagining a four-dimensional *space* that is "homogeneous," where every direction is like every other direction, where we can pick up a box and set it down so that no matter which three of the four directions we see, no one is distinguishable from any other.

Curiously, each of the three views used by magnetic resonance imaging
has appeared as a device in lower-dimensional science-fiction
allegories. *Flatland* encourages us to explore an axial
view of a two-dimensional universe. In his *An Episode of
Flatland*, also written in 1884, C. Howard Hinton, a
contemporary and probable inspiration of Abbott, proposed a race
of two-dimensional creatures from a *sagittal* viewpoint,
right- and left-handed triangular beings living on the outside
of a disc. Dionys Burger, in writing his 1964 sequel,
*Sphereland*, described symmetrical figures in a
*coronal* view. In our own day, A. K. Dewdney used a
mixture of sagittal and coronal views in his modern allegory,
*The Planiverse*. Each approach has its particular
geometrical features, and each suggests its own questions.

Slicing Basic Three-Dimensional Shapes | ||

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