A Medley of Potpourri

Thursday, August 31, 2023

Four-dimensional space

From Wikipedia, the free encyclopedia
https://en.wikipedia.org/wiki/Four-dimensional_space

Animation of a transforming tesseract or 4-cube — The 4D equivalent of a cube is known as a tesseract, seen rotating here in four-dimensional space, yet projected into two dimensions for display.

Four-dimensional space (4D) is the mathematical extension of the concept of three-dimensional space (3D). Three-dimensional space is the simplest possible abstraction of the observation that one needs only three numbers, called dimensions, to describe the sizes or locations of objects in the everyday world. For example, the volume of a rectangular box is found by measuring and multiplying its length, width, and height (often labeled $x$ , $y$ , and $z$ ). This concept of ordinary space is called Euclidean space because it corresponds to Euclid's geometry, which was originally abstracted from the spatial experiences of everyday life.

The idea of adding a fourth dimension appears in Jean le Rond d'Alembert's "Dimensions". published in 1754, but the mathematics of more than three dimensions only emerged in the 19th century. The general concept of Euclidean space with any number of dimensions was fully developed by the Swiss mathematician Ludwig Schläfli before 1853. Schläfli's work received little attention during his lifetime and was published only posthumously, in 1901, but meanwhile the fourth Euclidean dimension was rediscovered by others. In 1880 Charles Howard Hinton popularized it in an essay, "What is the Fourth Dimension?", in which he explained the concept of a "four-dimensional cube" with a step-by-step generalization of the properties of lines, squares, and cubes. The simplest form of Hinton's method is to draw two ordinary 3D cubes in 2D space, one encompassing the other, separated by an "unseen" distance, and then draw lines between their equivalent vertices. This can be seen in the accompanying animation whenever it shows a smaller inner cube inside a larger outer cube. The eight lines connecting the vertices of the two cubes in this case represent a single direction in the "unseen" fourth dimension.

Higher-dimensional spaces (greater than three) have since become one of the foundations for formally expressing modern mathematics and physics. Large parts of these topics could not exist in their current forms without using such spaces. Einstein's theory of relativity is formulated in 4D space, although not in a Euclidean 4D space. Einstein's concept of spacetime has a Minkowski structure based on a non-Euclidean geometry with three spatial dimensions and one temporal dimension, rather than the four symmetric spatial dimensions of Schläfli's Euclidean 4D space.

Single locations in Euclidean 4D space can be given as vectors or n-tuples, i.e., as ordered lists of numbers such as $(x, y, z, w)$ . It is only when such locations are linked together into more complicated shapes that the full richness and geometric complexity of higher-dimensional spaces emerge. A hint of that complexity can be seen in the accompanying 2D animation of one of the simplest possible regular 4D objects, the tesseract, which is analogous to the 3D cube.

History

Lagrange wrote in his Mécanique analytique (published 1788, based on work done around 1755) that mechanics can be viewed as operating in a four-dimensional space— three dimensions of space, and one of time. As early as 1827, Möbius realized that a fourth spatial dimension would allow a three-dimensional form to be rotated onto its mirror-image. The general concept of Euclidean space with any number of dimensions was fully developed by the Swiss mathematician Ludwig Schläfli in the mid-19th century, at a time when Cayley, Grassman and Möbius were the only other people who had ever conceived the possibility of geometry in more than three dimensions. By 1853 Schläfli had discovered all the regular polytopes that exist in higher dimensions, including the four-dimensional analogs of the Platonic solids.

An arithmetic of four spatial dimensions, called quaternions, was defined by William Rowan Hamilton in 1843. This associative algebra was the source of the science of vector analysis in three dimensions as recounted by Michael J. Crowe in A History of Vector Analysis. Soon after, tessarines and coquaternions were introduced as other four-dimensional algebras over R. In 1886, Victor Schlegel described his method of visualizing four-dimensional objects with Schlegel diagrams.

One of the first popular expositors of the fourth dimension was Charles Howard Hinton, starting in 1880 with his essay What is the Fourth Dimension?, published in the Dublin University magazine. He coined the terms tesseract, ana and kata in his book A New Era of Thought and introduced a method for visualizing the fourth dimension using cubes in the book Fourth Dimension. Hinton's ideas inspired a fantasy about a "Church of the Fourth Dimension" featured by Martin Gardner in his January 1962 "Mathematical Games column" in Scientific American.

Higher dimensional non-Euclidean spaces were put on a firm footing by Bernhard Riemann's 1854 thesis, Über die Hypothesen welche der Geometrie zu Grunde liegen, in which he considered a "point" to be any sequence of coordinates $(x 1, ..., x n)$ . In 1908, Hermann Minkowski presented a paper consolidating the role of time as the fourth dimension of spacetime, the basis for Einstein's theories of special and general relativity. But the geometry of spacetime, being non-Euclidean, is profoundly different from that explored by Schläfli and popularised by Hinton. The study of Minkowski space required Riemann's mathematics which is quite different from that of four-dimensional Euclidean space, and so developed along quite different lines. This separation was less clear in the popular imagination, with works of fiction and philosophy blurring the distinction, so in 1973 H. S. M. Coxeter felt compelled to write:

Little, if anything, is gained by representing the fourth Euclidean dimension as time. In fact, this idea, so attractively developed by H. G. Wells in The Time Machine, has led such authors as John William Dunne (An Experiment with Time) into a serious misconception of the theory of Relativity. Minkowski's geometry of space-time is not Euclidean, and consequently has no connection with the present investigation.
— H. S. M. Coxeter, Regular Polytopes

Vectors

Mathematically, a four-dimensional space is a space with four spatial dimensions, that is a space that needs four parameters to specify a point in it. For example, a general point might have position vector $a$ , equal to

a = (\begin{matrix} a_{1} \\ a_{2} \\ a_{3} \\ a_{4} \end{matrix}) .

This can be written in terms of the four standard basis vectors $(e 1, e 2, e 3, e 4)$ , given by

e_{1} = (\begin{matrix} 1 \\ 0 \\ 0 \\ 0 \end{matrix}); e_{2} = (\begin{matrix} 0 \\ 1 \\ 0 \\ 0 \end{matrix}); e_{3} = (\begin{matrix} 0 \\ 0 \\ 1 \\ 0 \end{matrix}); e_{4} = (\begin{matrix} 0 \\ 0 \\ 0 \\ 1 \end{matrix}),

so the general vector $a$ is

a = a_{1} e_{1} + a_{2} e_{2} + a_{3} e_{3} + a_{4} e_{4} .

Vectors add, subtract and scale as in three dimensions.

The dot product of Euclidean three-dimensional space generalizes to four dimensions as

a \cdot b = a_{1} b_{1} + a_{2} b_{2} + a_{3} b_{3} + a_{4} b_{4} .

It can be used to calculate the norm or length of a vector,

| a | = \sqrt{a \cdot a} = \sqrt{a_{1}^{2} + a_{2}^{2} + a_{3}^{2} + a_{4}^{2}},

and calculate or define the angle between two non-zero vectors as

θ = \arccos \frac{a \cdot b}{| a | | b |} .

Minkowski spacetime is four-dimensional space with geometry defined by a non-degenerate pairing different from the dot product:

a \cdot b = a_{1} b_{1} + a_{2} b_{2} + a_{3} b_{3} - a_{4} b_{4} .

As an example, the distance squared between the points $(0,0,0,0)$ and $(1,1,1,0)$ is 3 in both the Euclidean and Minkowskian 4-spaces, while the distance squared between $(0,0,0,0)$ and $(1,1,1,1)$ is 4 in Euclidean space and 2 in Minkowski space; increasing $b 4$ decreases the metric distance. This leads to many of the well-known apparent "paradoxes" of relativity.

The cross product is not defined in four dimensions. Instead, the exterior product is used for some applications, and is defined as follows:

\begin{aligned} a \land b = (a_{1} b_{2} - a_{2} b_{1}) e_{12} + (a_{1} b_{3} - a_{3} b_{1}) e_{13} + (a_{1} b_{4} - a_{4} b_{1}) e_{14} + (a_{2} b_{3} - a_{3} b_{2}) e_{23} \\ + (a_{2} b_{4} - a_{4} b_{2}) e_{24} + (a_{3} b_{4} - a_{4} b_{3}) e_{34} . \end{aligned}

This is bivector valued, with bivectors in four dimensions forming a six-dimensional linear space with basis $(e 12, e 13, e 14, e 23, e 24, e 34)$ . They can be used to generate rotations in four dimensions.

Orthogonality and vocabulary

In the familiar three-dimensional space of daily life, there are three coordinate axes—usually labeled $x$ , $y$ , and $z$ —with each axis orthogonal (i.e. perpendicular) to the other two. The six cardinal directions in this space can be called up, down, east, west, north, and south. Positions along these axes can be called altitude, longitude, and latitude. Lengths measured along these axes can be called height, width, and depth.

Comparatively, four-dimensional space has an extra coordinate axis, orthogonal to the other three, which is usually labeled $w$ . To describe the two additional cardinal directions, Charles Howard Hinton coined the terms ana and kata, from the Greek words meaning "up toward" and "down from", respectively.

As mentioned above, Hermann Minkowski exploited the idea of four dimensions to discuss cosmology including the finite velocity of light. In appending a time dimension to three-dimensional space, he specified an alternative perpendicularity, hyperbolic orthogonality. This notion provides his four-dimensional space with a modified simultaneity appropriate to electromagnetic relations in his cosmos. Minkowski's world overcame problems associated with the traditional absolute space and time cosmology previously used in a universe of three space dimensions and one time dimension.

Geometry

The geometry of four-dimensional space is much more complex than that of three-dimensional space, due to the extra degree of freedom.

Just as in three dimensions there are polyhedra made of two dimensional polygons, in four dimensions there are polychora made of polyhedra. In three dimensions, there are 5 regular polyhedra known as the Platonic solids. In four dimensions, there are 6 convex regular 4-polytopes, the analogs of the Platonic solids. Relaxing the conditions for regularity generates a further 58 convex uniform 4-polytopes, analogous to the 13 semi-regular Archimedean solids in three dimensions. Relaxing the conditions for convexity generates a further 10 nonconvex regular 4-polytopes.

Regular polytopes in four dimensions
(Displayed as orthogonal projections in each Coxeter plane of symmetry)
A₄, [3,3,3]	B₄, [4,3,3]		F₄, [3,4,3]	H₄, [5,3,3]
5-cell {3,3,3}	tesseract {4,3,3}	16-cell {3,3,4}	24-cell {3,4,3}	600-cell {3,3,5}	120-cell {5,3,3}

In three dimensions, a circle may be extruded to form a cylinder. In four dimensions, there are several different cylinder-like objects. A sphere may be extruded to obtain a spherical cylinder (a cylinder with spherical "caps", known as a spherinder), and a cylinder may be extruded to obtain a cylindrical prism (a cubinder). The Cartesian product of two circles may be taken to obtain a duocylinder. All three can "roll" in four-dimensional space, each with its properties.

In three dimensions, curves can form knots but surfaces cannot (unless they are self-intersecting). In four dimensions, however, knots made using curves can be trivially untied by displacing them in the fourth direction—but 2D surfaces can form non-trivial, non-self-intersecting knots in 4D space. Because these surfaces are two-dimensional, they can form much more complex knots than strings in 3D space can. The Klein bottle is an example of such a knotted surface. Another such surface is the real projective plane.

Hypersphere

Stereographic projection of a Clifford torus: the set of points $(cos(a), sin(a), cos(b), sin(b))$ , which is a subset of the 3-sphere.

The set of points in Euclidean 4-space having the same distance $R$ from a fixed point $P 0$ forms a hypersurface known as a 3-sphere. The hyper-volume of the enclosed space is:

V = \begin{matrix} \frac{1}{2} \end{matrix} π^{2} R^{4}

This is part of the Friedmann–Lemaître–Robertson–Walker metric in General relativity where $R$ is substituted by function $R (t)$ with $t$ meaning the cosmological age of the universe. Growing or shrinking $R$ with time means expanding or collapsing universe, depending on the mass density inside.

Four-dimensional perception in humans

Research using virtual reality finds that humans, despite living in a three-dimensional world, can, without special practice, make spatial judgments about line segments embedded in four-dimensional space, based on their length (one-dimensional) and the angle (two-dimensional) between them. The researchers noted that "the participants in our study had minimal practice in these tasks, and it remains an open question whether it is possible to obtain more sustainable, definitive, and richer 4D representations with increased perceptual experience in 4D virtual environments". In another study, the ability of humans to orient themselves in 2D, 3D, and 4D mazes has been tested. Each maze consisted of four path segments of random length and connected with orthogonal random bends, but without branches or loops (i.e. actually labyrinths). The graphical interface was based on John McIntosh's free 4D Maze game. The participating persons had to navigate through the path and finally estimate the linear direction back to the starting point. The researchers found that some of the participants were able to mentally integrate their path after some practice in 4D (the lower-dimensional cases were for comparison and for the participants to learn the method).

However, a 2020 review underlined how these studies are composed of a small subject sample and mainly of college students. It also pointed out other issues that future research has to resolve: elimination of artifacts (these could be caused, for example, by strategies to resolve the required task that don't use 4D representation/4D reasoning and feedback given by researchers to speed up the adaptation process) and analysis on inter-subject variability (if 4D perception is possible, its acquisition could be limited to a subset of humans, to a specific critical period, or to people's attention or motivation). Furthermore, it is needed to determine if there is a more appropriate way to project the 4-dimension (because there are no restrictions on how the 4-dimension can be projected). Researchers also hypothesized that human acquisition of 4D perception could result in the activation of brain visual areas and entorhinal cortex. If so they suggest that it could be used as a strong indicator of 4D space perception acquisition. Authors also suggested using a variety of different neural network architectures (with different a priori assumptions) to understand which ones are or are not able to learn.

Dimensional analogy

To understand the nature of four-dimensional space, a device called dimensional analogy is commonly employed. Dimensional analogy is the study of how ( $n - 1$ ) dimensions relate to $n$ dimensions, and then inferring how $n$ dimensions would relate to ( $n + 1$ ) dimensions.

The dimensional analogy was used by Edwin Abbott Abbott in the book Flatland, which narrates a story about a square that lives in a two-dimensional world, like the surface of a piece of paper. From the perspective of this square, a three-dimensional being has seemingly god-like powers, such as ability to remove objects from a safe without breaking it open (by moving them across the third dimension), to see everything that from the two-dimensional perspective is enclosed behind walls, and to remain completely invisible by standing a few inches away in the third dimension.

By applying dimensional analogy, one can infer that a four-dimensional being would be capable of similar feats from the three-dimensional perspective. Rudy Rucker illustrates this in his novel Spaceland, in which the protagonist encounters four-dimensional beings who demonstrate such powers.

Cross-sections

As a three-dimensional object passes through a two-dimensional plane, two-dimensional beings in this plane would only observe a cross-section of the three-dimensional object within this plane. For example, if a sphere passed through a sheet of paper, beings in the paper would see first a single point. A circle gradually grows larger, until it reaches the diameter of the sphere, and then gets smaller again, until it shrinks to a point and disappears. The 2D beings would not see a circle in the same way as three-dimensional beings do; rather, they only see a one-dimensional projection of the circle on their 1D "retina". Similarly, if a four-dimensional object passed through a three-dimensional (hyper) surface, one could observe a three-dimensional cross-section of the four-dimensional object. For example, a hypersphere would appear first as a point, then as a growing sphere (until it reaches the "hyperdiameter" of the hypersphere), with the sphere then shrinking to a single point and then disappearing. This means of visualizing aspects of the fourth dimension was used in the novel Flatland and also in several works of Charles Howard Hinton. And, in the same way, three-dimensional beings (such as humans with a 2D retina) can see all the sides and the insides of a 2D shape simultaneously, a 4D being could see all faces and the inside of a 3D shape at once with their 3D retina.

Projections

A useful application of dimensional analogy in visualizing higher dimensions is in projection. A projection is a way of representing an n-dimensional object in $n - 1$ dimensions. For instance, computer screens are two-dimensional, and all the photographs of three-dimensional people, places, and things are represented in two dimensions by projecting the objects onto a flat surface. By doing this, the dimension orthogonal to the screen (depth) is removed and replaced with indirect information. The retina of the eye is also a two-dimensional array of receptors but the brain can perceive the nature of three-dimensional objects by inference from indirect information (such as shading, foreshortening, binocular vision, etc.). Artists often use perspective to give an illusion of three-dimensional depth to two-dimensional pictures. The shadow, cast by a fictitious grid model of a rotating tesseract on a plane surface, as shown in the figures, is also the result of projections.

Similarly, objects in the fourth dimension can be mathematically projected to the familiar three dimensions, where they can be more conveniently examined. In this case, the 'retina' of the four-dimensional eye is a three-dimensional array of receptors. A hypothetical being with such an eye would perceive the nature of four-dimensional objects by inferring four-dimensional depth from indirect information in the three-dimensional images in its retina.

The perspective projection of three-dimensional objects into the retina of the eye introduces artifacts such as foreshortening, which the brain interprets as depth in the third dimension. In the same way, perspective projection from four dimensions produces similar foreshortening effects. By applying dimensional analogy, one may infer four-dimensional "depth" from these effects.

As an illustration of this principle, the following sequence of images compares various views of the three-dimensional cube with analogous projections of the four-dimensional tesseract into three-dimensional space.

Cube	Tesseract	Description
		The image on the left is a cube viewed face-on. The analogous viewpoint of the tesseract in 4 dimensions is the cell-first perspective projection, shown on the right. One may draw an analogy between the two: just as the cube projects to a square, the tesseract projects to a cube. Note that the other 5 faces of the cube are not seen here. They are obscured by the visible face. Similarly, the other 7 cells of the tesseract are not seen here because they are obscured by the visible cell.
		The image on the left shows the same cube viewed edge-on. The analogous viewpoint of a tesseract is the face-first perspective projection, shown on the right. Just as the edge-first projection of the cube consists of two trapezoids, the face-first projection of the tesseract consists of two frustums. The nearest edge of the cube in this viewpoint is the one lying between the red and green faces. Likewise, the nearest face of the tesseract is the one lying between the red and green cells.
		On the left is the cube viewed corner-first. This is analogous to the edge-first perspective projection of the tesseract, shown on the right. Just as the cube's vertex-first projection consists of 3 deltoids surrounding a vertex, the tesseract's edge-first projection consists of 3 hexahedral volumes surrounding an edge. Just as the nearest vertex of the cube is the one where the three faces meet, the nearest edge of the tesseract is the one in the center of the projection volume, where the three cells meet.
		A different analogy may be drawn between the edge-first projection of the tesseract and the edge-first projection of the cube. The cube's edge-first projection has two trapezoids surrounding an edge, while the tesseract has three hexahedral volumes surrounding an edge.
		On the left is the cube viewed corner-first. The vertex-first perspective projection of the tesseract is shown on the right. The cube's vertex-first projection has three tetragons surrounding a vertex, but the tesseract's vertex-first projection has four hexahedral volumes surrounding a vertex. Just as the nearest corner of the cube is the one lying at the center of the image, so the nearest vertex of the tesseract lies not on the boundary of the projected volume, but at its center inside, where all four cells meet. Only three of the cube's six faces can be seen here, because the other three faces lie behind these three faces, on the opposite side of the cube. Similarly, only four of the tesseract's eight cells can be seen here; the remaining four lie behind these four in the fourth direction, on the far side of the tesseract.

Shadows

A concept closely related to projection is the casting of shadows.

If a light is shone on a three-dimensional object, a two-dimensional shadow is cast. By dimensional analogy, light shone on a two-dimensional object in a two-dimensional world would cast a one-dimensional shadow, and light on a one-dimensional object in a one-dimensional world would cast a zero-dimensional shadow, that is, a point of non-light. Going the other way, one may infer that light shining on a four-dimensional object in a four-dimensional world would cast a three-dimensional shadow.

If the wireframe of a cube is lit from above, the resulting shadow on a flat two-dimensional surface is a square within a square with the corresponding corners connected. Similarly, if the wireframe of a tesseract were lit from "above" (in the fourth dimension), its shadow would be that of a three-dimensional cube within another three-dimensional cube suspended in midair (a "flat" surface from a four-dimensional perspective). (Note that, technically, the visual representation shown here is a two-dimensional image of the three-dimensional shadow of the four-dimensional wireframe figure.)

Bounding volumes

The dimensional analogy also helps in inferring basic properties of objects in higher dimensions. For example, two-dimensional objects are bounded by one-dimensional boundaries: a square is bounded by four edges. Three-dimensional objects are bounded by two-dimensional surfaces: a cube is bounded by 6 square faces. By applying dimensional analogy, one may infer that a four-dimensional cube, known as a tesseract, is bounded by three-dimensional volumes. And indeed, this is the case: mathematics shows that the tesseract is bounded by 8 cubes. Knowing this is key to understanding how to interpret a three-dimensional projection of the tesseract. The boundaries of the tesseract project to volumes in the image, not merely two-dimensional surfaces.

Visual scope

People have a spatial self-perception as beings in a three-dimensional space, but are visually restricted by one less dimension: the eye sees the world as a projection to two dimensions, on the surface of the retina. Assuming a four-dimensional being were able to see the world in projections to a hypersurface, also just one dimension less, i.e., to three dimensions, it would be able to see, e.g., all six faces of an opaque box simultaneously, and in fact, what is inside the box at the same time, just as people can see all four sides and simultaneously the interior of a rectangle on a piece of paper. The being would be able to discern all points in a 3-dimensional subspace simultaneously, including the inner structure of solid 3-dimensional objects, things obscured from human viewpoints in three dimensions on two-dimensional projections. Brains receive images in two dimensions and use reasoning to help picture three-dimensional objects.

Limitations

Reasoning by analogy from familiar lower dimensions can be an excellent intuitive guide, but care must be exercised not to accept results that are not more rigorously tested. For example, consider the formulas for the area enclosed by a circle in two dimensions ( $A = π r^{2}$ ) and the volume enclosed by a sphere in three dimensions ( $V = \frac{4}{3} π r^{3}$ ). One might guess that the volume enclosed by the sphere in four-dimensional space is a rational multiple of $π r^{4}$ , but the correct volume is $\frac{π^{2}}{2} r^{4}$ . The volume in all dimensions is computable from a recurrence relation connecting dimension $n$ to dimension $n - 2$ .

In culture

In art

An illustration from Jouffret's *Traité élémentaire de géométrie à quatre dimensions*. The book, which influenced Picasso, was given to him by Princet.

New possibilities opened up by the concept of four-dimensional space (and difficulties involved in trying to visualize it) helped inspire many modern artists in the first half of the twentieth century. Early Cubists, Surrealists, Futurists, and abstract artists took ideas from higher-dimensional mathematics and used them to radically advance their work.

In literature

Science fiction texts often mention the concept of "dimension" when referring to parallel or alternate universes or other imagined planes of existence. This usage is derived from the idea that to travel to parallel/alternate universes/planes of existence one must travel in a direction/dimension besides the standard ones. In effect, the other universes/planes are just a small distance away from our own, but the distance is in a fourth (or higher) spatial (or non-spatial) dimension, not the standard ones.

One of the most heralded science fiction stories regarding true geometric dimensionality, and often recommended as a starting point for those just starting to investigate such matters, is the 1884 novella Flatland by Edwin A. Abbott. Isaac Asimov, in his foreword to the Signet Classics 1984 edition, described Flatland as "The best introduction one can find into the manner of perceiving dimensions."

The idea of other dimensions was incorporated into many early science fiction stories, appearing prominently, for example, in Miles J. Breuer's The Appendix and the Spectacles (1928) and Murray Leinster's The Fifth-Dimension Catapult (1931); and appeared irregularly in science fiction by the 1940s. Classic stories involving other dimensions include Robert A. Heinlein's —And He Built a Crooked House (1941), in which a California architect designs a house based on a three-dimensional projection of a tesseract; Alan E. Nourse's Tiger by the Tail and The Universe Between (both 1951); and The Ifth of Oofth (1957) by Walter Tevis. Another reference is Madeleine L'Engle's novel A Wrinkle In Time (1962), which uses the fifth dimension as a way of "tesseracting the universe" or "folding" space to move across it quickly. The fourth and fifth dimensions are also key components of the book The Boy Who Reversed Himself by William Sleator.

In philosophy

Immanuel Kant, in 1783, wrote: "That everywhere space (which is not itself the boundary of another space) has three dimensions and that space, in general, cannot have more dimensions is based on the proposition that not more than three lines can intersect at right angles in one point. This proposition cannot at all be shown from concepts, but rests immediately on intuition and indeed on pure intuition a priori because it is apodictically (demonstrably) certain."

"Space has Four Dimensions" is a short story published in 1846 by German philosopher and experimental psychologist Gustav Fechner under the pseudonym "Dr. Mises". The protagonist in the tale is a shadow who is aware of and able to communicate with other shadows, but who is trapped on a two-dimensional surface. According to Fechner, this "shadow-man" would conceive of the third dimension as being one of time. The story bears a strong similarity to the "Allegory of the Cave" presented in Plato's The Republic (c. 380 BC).

Simon Newcomb wrote an article for the Bulletin of the American Mathematical Society in 1898 entitled "The Philosophy of Hyperspace". Linda Dalrymple Henderson coined the term "hyperspace philosophy", used to describe writing that uses higher dimensions to explore metaphysical themes, in her 1983 thesis about the fourth dimension in early-twentieth-century art. Examples of "hyperspace philosophers" include Charles Howard Hinton, the first writer, in 1888, to use the word "tesseract"; and the Russian esotericist P. D. Ouspensky.

Wednesday, August 30, 2023

Flatland

From Wikipedia, the free encyclopedia

https://en.wikipedia.org/wiki/Flatland

Flatland: A Romance of Many Dimensions
The cover to Flatland, first edition
Author	Edwin A. Abbott
Illustrator	Edwin A. Abbott
Country	England
Genre	Mathematical fiction
Publisher	Seeley & Co.
Publication date	1884
Pages	96
OCLC	2306280
LC Class	QA699

Flatland: A Romance of Many Dimensions is a satirical novella by the English schoolmaster Edwin Abbott Abbott, first published in 1884 by Seeley & Co. of London. Written pseudonymously by "A Square", the book used the fictional two-dimensional world of Flatland to comment on the hierarchy of Victorian culture, but the novella's more enduring contribution is its examination of dimensions.

Several films have been made from the story, including the feature film Flatland (2007). Other efforts have been short or experimental films, including one narrated by Dudley Moore and the short films Flatland: The Movie (2007) and Flatland 2: Sphereland (2012).

Plot

The story describes a two-dimensional world inhabited by geometric figures; women are line segments, while men are polygons with various numbers of sides. The narrator is a square, a member of the caste of gentlemen and professionals, who guides the readers through some of the implications of life in two dimensions. The first half of the story goes through the practicalities of existing in a two-dimensional universe, as well as a history leading up to the year 1999 on the eve of the 3rd Millennium.

On New Year's Eve, the Square dreams of a visit to a one-dimensional world, "Lineland", inhabited by men, consisting of lines, while the women consisted of "lustrous points". These points and lines are unable to see the Square as anything other than a set of points on a line. Thus, the Square attempts to convince the realm's monarch of a second dimension but cannot do so. In the end, the monarch of Lineland tries to kill the Square rather than tolerate him any further.

Following this vision, the Square is visited by a sphere. Similar to the "points" in Lineland, he is unable to see the three-dimensional object as anything other than a circle (more precisely, a disk). The Sphere then levitates up and down through Flatland, allowing the Square to see the circle expand and contract between great circle and small circles. The Sphere then tries further to convince the Square of the third dimension by dimensional analogies (a point becomes a line, a line becomes a square). The Square is still unable to comprehend the third dimension, so the Sphere resorts to deeds: he gives info about the "insides" of the house, moves a cup through the third dimension, and even goes inside the Square for a bit. Still unable to comprehend 3D, the Sphere takes the Square to the third dimension, Spaceland. This Sphere visits Flatland at the turn of each millennium to introduce a new apostle to the idea of a third dimension in the hope of eventually educating the population of Flatland. From the safety of Spaceland, they can oversee the leaders of Flatland, acknowledging the Sphere's existence and prescribing the silencing. After this proclamation is made, many witnesses are massacred or imprisoned (according to caste), including the Square's brother.

After the Square's mind is opened to new dimensions, he tries to convince the Sphere of the theoretical possibility of the existence of a fourth dimension and higher spatial dimensions. Still, the Sphere returns his student to Flatland in disgrace.

The Square then has a dream in which the Sphere revisits him, this time to introduce him to a zero-dimensional space, Pointland, of whom the Point (sole inhabitant, monarch, and universe in one) perceives any communication as a thought originating in his own mind (cf. Solipsism):

"You see," said my Teacher, "how little your words have done. So far as the Monarch understands them at all, he accepts them as his own – for he cannot conceive of any other except himself – and plumes himself upon the variety of Its Thought as an instance of creative Power. Let us leave this god of Pointland to the ignorant fruition of his omnipresence and omniscience: nothing that you or I can do can rescue him from his self-satisfaction."
— the Sphere

The Square recognises the identity of the ignorance of the monarchs of Pointland and Lineland with his own (and the Sphere's) previous ignorance of the existence of higher dimensions. Once returned to Flatland, the Square cannot convince anyone of Spaceland's existence, especially after official decrees are announced that anyone preaching the existence of three dimensions will be imprisoned (or executed, depending on caste). For example, he tries to convince his relative of the third dimension but cannot move a square "upward," as opposed to forward or sideways. Eventually, the Square himself is imprisoned for just this reason, with only occasional contact with his brother, who is imprisoned in the same facility. He cannot convince his brother, even after all they have both seen. Seven years after being imprisoned, A Square writes out the book Flatland as a memoir, hoping to keep it as posterity for a future generation that can see beyond their two-dimensional existence.

Social elements

Men are portrayed as polygons whose social status is determined by their regularity and the number of their sides, with a Circle considered the "perfect" shape. Women are lines, quite fragile but also dangerous, as they can disappear from view and possibly stab someone. To prevent this they are required by law to sound a "peace-cry" while moving about and to use separate doors from men.

In the world of Flatland, classes are distinguished by the "Art of Hearing", the "Art of Feeling", and the "Art of Sight Recognition". Classes can be distinguished by the sound of one's voice, but the lower classes have more developed vocal organs, enabling them to feign the voice of a Polygon or even a Circle. Feeling, practised by the lower classes and women, determines the configuration of a person by feeling one of its angles. The "Art of Sight Recognition", practised by the upper classes, is aided by "Fog", which allows an observer to determine the depth of an object. With this, polygons with sharp angles relative to the observer will fade more rapidly than polygons with more gradual angles. Colour of any kind was banned in Flatland after Isosceles workers painted themselves to impersonate noble Polygons. The Square describes these events, and the ensuing class war at length.

The population of Flatland can "evolve" through the "Law of Nature", which states: "a male child shall have one more side than his father, so that each generation shall rise (as a rule) one step in the scale of development and nobility. Thus the son of a Square is a Pentagon, the son of a Pentagon, a Hexagon; and so on".

This rule is not the case when dealing with Isosceles Triangles (Soldiers and Workmen) with only two congruent sides. The smallest angle of an Isosceles Triangle gains 30 arc minutes (half a degree) each generation. Additionally, the rule does not seem to apply to many-sided Polygons. For example, the sons of several hundred-sided Polygons will often develop 50 or more sides more than their parents. Furthermore, the angle of an Isosceles Triangle or the number of sides of a (regular) Polygon may be altered during life by deeds or surgical adjustments.

An Equilateral Triangle is a member of the craftsman class. Squares and Pentagons are the "gentlemen" class, as doctors, lawyers, and other professions. Hexagons are the lowest rank of nobility, all the way up to (near) Circles, who make up the priest class. The higher-order Polygons have much less of a chance of producing sons, preventing Flatland from being overcrowded with noblemen.

Apart from Isosceles Triangles, only regular Polygons are considered until chapter seven of the book when the issue of irregularity, or physical deformity is brought up. In a two-dimensional world, a regular polygon can be identified by a single angle and/or vertex. To maintain social cohesion, irregularity is to be abhorred, with moral irregularity and criminality cited, "by some" (in the book), as inevitable additional deformities, a sentiment with which the Square concurs. If the error of deviation is above a stated amount, the irregular Polygon faces euthanasia; if below, he becomes the lowest rank of civil servant. An irregular Polygon is not destroyed at birth, but allowed to develop to see if the irregularity can be "cured" or reduced. If the deformity remains, the irregular is "painlessly and mercifully consumed."

As social satire

In Flatland, Abbott describes a society rigidly divided into classes. Social ascent is the main aspiration of its inhabitants, apparently granted to everyone but strictly controlled by the top of the hierarchy. Freedom is despised and the laws are cruel. Innovators are imprisoned or suppressed. Members of lower classes who are intellectually valuable, and potential leaders of riots, are either killed or promoted to the higher classes. Every attempt for change is considered dangerous and harmful. This world is not prepared to receive "revelations from another world". The satirical part is mainly concentrated in the first part of the book, "This World", which describes Flatland. The main points of interest are the Victorian concept of women's roles in the society and in the class-based hierarchy of men. Abbott has been accused of misogyny due to his portrayal of women in Flatland. In his Preface to the Second and Revised Edition, 1884, he answers such critics by emphasizing that the description of women was satirizing the viewpoints held, stating that the Square:

was writing as a Historian, he has identified himself (perhaps too closely) with the views generally adopted by Flatland and (as he has been informed) even by Spaceland, Historians; in whose pages (until very recent times) the destinies of Women and of the masses of mankind have seldom been deemed worthy of mention and never of careful consideration.

Critical reception

Flatland did not have much success when published, although it was not entirely ignored. In the entry on Edwin Abbott in the Dictionary of National Biography for persons who died in the period of 1922 to 1930, Flatland was not even mentioned.

The book was discovered again after Albert Einstein's general theory of relativity was published, which brought to prominence the concept of a fourth dimension. Flatland was mentioned in a letter by William Garnett entitled "Euclid, Newton and Einstein" published in Nature on 12 February 1920. In this letter, Abbott is depicted, in a sense, as a prophet due to his intuition of the importance of time to explain certain phenomena:

Some thirty or more years ago a little jeu d'esprit was written by Dr. Edwin Abbott entitled Flatland. At the time of its publication it did not attract as much attention as it deserved... If there is motion of our three-dimensional space relative to the fourth dimension, all the changes we experience and assign to the flow of time will be due simply to this movement, the whole of the future as well as the past always existing in the fourth dimension.

The Oxford Dictionary of National Biography subsequently revised his biography, and as of 2020 it states that [Abbott] "is most remembered as the author of Flatland: A Romance of Many Dimensions".

Adaptations and parodies

Numerous imitations or sequels to Flatland have been created. Examples include:

Films and TV

Flatland (1965), an animated short film based on the novella, was directed by Eric Martin and based on an idea by John Hubley.
Flatlandia (1982), an Italian animated short film, directed by Michele Emmer.
Flatland (2007), a 98-minute animated independent feature film version directed by Ladd Ehlinger Jr., updates the satire from Victorian England to the modern-day United States.
Flatland: The Movie (2007), by Dano Johnson and Jeffrey Travis, is a 34-minute animated educational film. Its sequel was Flatland 2: Sphereland (2012), inspired by the novel Sphereland by Dionys Burger.
In the Star Trek: The Next Generation episode "The Loss", the USS Enterprise-D becomes trapped within a field of two-dimensional lifeforms.

Literature

Books and short stories inspired by Flatland include:

An Episode on Flatland: Or How a Plain Folk Discovered the Third Dimension by Charles Howard Hinton (1907)
The Dot and the Line: A Romance in Lower Mathematics by Norton Juster (1963)
Sphereland by Dionys Burger (1965)
The Incredible Umbrella by Marvin Kaye (1980)
"Message Found in a Copy of Flatland" by Rudy Rucker (1983)
The Planiverse by A. K. Dewdney (1984)
Flatterland by Ian Stewart (2001)
Spaceland by Rudy Rucker (2002)
VAS: An Opera in Flatland (2002) by Steve Tomasula, which uses the two-dimensional world to critique contemporary society

In popular culture

Physicists and science popularizers Carl Sagan and Stephen Hawking have both commented on and postulated about the effects of Flatland. Sagan recreates the thought experiment as a set-up to discussing the possibilities of higher dimensions of the physical universe in both the book and television series Cosmos, whereas Hawking notes the peculiarity of life in two-dimensional space, as any inhabitants would necessarily be unable to digest their own food. (This concept is parodied in the below-described episode of Futurama. The protagonists attempt to eat Flatland food but it falls out immediately. The native organisms in Flatland absorb food somewhat like amoeba.)
The League of Extraordinary Gentlemen, Volume 2, issue 3, in chapter 3 of the series of writings New Traveller's Almanac, it is mentioned that in an unknown basement of New York, Flatland was discovered by a mathematician.
In the "2-D Blacktop" episode of the animated science fiction TV comedy series Futurama (season 7, episode 15, originally broadcast June 19, 2013), two spaceships moving at relativistic speeds crash head on and are compressed together into a flat disk. They meet natives of the realm, who chase after them when the concept of a third dimension is brought up.
In David Foster Wallace's novel Infinite Jest (1996), it is briefly mentioned that students from the Enfield Tennis Academy could be seen studying and highlighting copies of Flatland on the bus.
Flatland features in The Big Bang Theory episode "The Psychic Vortex", when Sheldon Cooper declares it one of his favourite imaginary places to visit.
On the series The Orville, episode "New Dimensions", after entering a region of two-dimensional space Captain Ed Mercer references Flatland and its theme of social hierarchy.
In the Sons of Anarchy episode "Straw", Clay Morrow is lounging on a cot in a private cell in county jail when he first meets retired U.S. Marshal Lee Toric. Morrow half-ignores Toric while keeping his eyes on a copy of Flatland.
In Gravity Falls it is implied by the main antagonist Bill Cipher that he originates from a dimension very similar to Flatland.