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Visualizing Objects of Four-Dimensional Space:
From Flatland to the Hopf Fibration
Zamboj Michal
Department of Mathematics and Mathematical Education
Faculty of Education Charles University
M. D. Rettigov´
e 4, 116 39 Prague 1, Czech Republic
E-mail: michal.zamboj@pedf.cuni.cz
Abstract
One of the fundamental questions of a three-dimensional geometer is how to imagine a
four-dimensional object. And yet, he draws pictures of three-dimensional objects in the two-
dimensional paper. Moreover, would a two-dimensional geometer understand our sketches?
Based on analogies, we give an overview of methods of examination of four-dimensional ob-
jects. We emphasize visualization as the main element of perception of four-dimensional space.
For this purpose, we describe the double orthogonal projection of the four-dimensional space
onto two mutually perpendicular three-dimensional spaces as a generalization of the classical
Monge’s projection. In such a projection, we construct a four-dimensional playground for con-
venient synthetic creation of four-dimensional objects. All our constructions are easily acces-
sible with the interactive 3D modeling software GeoGebra. Furthermore, we apply the method
of projection to an intuitive investigation of various four-dimensional mathematical phenomena
– polytopes, four-dimensional quadrics, three-sphere and its stereographic projection, complex
plane, and the Hopf fibration.
fourth dimension, mathematical visualization, descriptive geometry, double orthogonal
projection
Primary 00A66; Secondary 51N05, 97G10
1 Introduction
This paper is a point on a line of papers about the fourth dimension. If the reader perceives a
paper as a flat piece of a plane, then this paper has two complex dimensions. Each complex
dimension may be generated by two real dimensions of this contribution, and hence we have
overall four dimensions: mathematical, visual, expository, narrative. Joining separately two
and two arbitrary dimensions, one can in complexity assume, for example, that the article is
about mathematical visualization of the fourth dimension with expository metaphores, or math-
ematical explanation through narrative visualizations of four-dimensional phenomena, etc. . ..
Summing up, this real complex contribution is a quaternionic point with an emphasis on the
visual dimension.
1
2 Four dimensions of four-dimensional visualization
2.1 Mathematical dimension
Since Ancient Greece, mathematicians (physicists, philosophers) have been denying, disprov-
ing, applying, denoting, generalizing, describing, accepting, defining, understanding, tirivializ-
ing the geometric concept of multidimensionality.
. . .three dimensions are all that there are .. .
Aristotle [84, Book I, Part I], 4th century BC
For example, in the context of antique geometry, we may observe a wonderful mystery in
the well-known Hero’s formula S4=ps(s−a)(s−b)(s−c)(see four-dimensional geo-
metric proof through decomposition in [19]), who, in his Metrika, 1st cenutry, called the fourth
power of magnitude as “dynamodynamis”, where dynamis is a square number (see [26]). The
knowledge of this formula is nowadays attributed to Archimedes ( [79, p. 84], [86, p.73]). In
geometry, Ptolemy proved, that there are at most three lines that are mutually perpendicular
(see [15]). Algebraic terminology for powers was, without geometric interpretation, adopted
and extented by Arabs, such as al-F´
ar´
ab´
ı, Abu’l-Vaf´
a, al-Khw´
arizm´
ı ( [26], [56, p.38]). ˇ
Ciˇ
zm´
ar
in [26] traced the linguistic relation from “alogos protos” = “first inexpressible” to denote
the fifth power by Michael Psellos, through translations in 15th and 16th century to “sur-
dum”=“deaf” →“solidum surdum” →“supersolidum”= “hyper solid” (c.f. [62, p. 3]).
Cajori in [15] assigned the phrase “4am dimensionem” to Oresme, who described extensions
(latitudes) of one (on a line), two (in a plane), three (in a space) quantities with some numeric
intensity (longitude) in Tractatus de configurationibus qualitatum et motuum, 1371 . Therefore,
Oresme is sometimes considered as the founder of the concept of function. In this case, one can
interpret intensity as a function of one, two or three extensions, creating with its value a planar
figure, a solid, or . . . , as same as Oresme, avoid it with infinitely many solids, see [15, 98],
and [26] for details.
“. . .the first power refers to a line, the square to a surface, the cube to a solid and that it
would be fatuous indeed for us to progress beyond, for the reason that it is contrary to nature.”
Cardano, [17, Part I], 1545, translation from [15]
. . .as same as if there were more than three dimensions, which is also against the nature.
Stifel [73, fol. 9], 1553
Symbolic representation played an important role in algebra, too. Vi`
ete in [90, fol. 6,7], 1591,
represented equations with vowels as unknowns and consonants as coefficients, while all the
terms had to be in the same dimension. Kvasz in [56, p.35] gives the following example:
B latus in A solidum - C quadratum in E planum equatur D quadrato-quadratum, is in our
notation ax3−by2=cwith all terms in the fourth dimension.
Another huge step was understanding a segment as a product of other segments, and hence not
to cross higher dimensions. Introducing analytic geometry, Descartes solved this in G´
eom´
etrie
[28], 1637 (see [56, pp. 37–40]).
“. . .as many dimensions as there are letters, but it will only be imaginary because in nature
we do not know of any quantity which has more than three dimensions.”
Ozanam, [67, p. 62], 1691, translation from [62, p. 3]
2
John Wallis in his Algebra distinguished algebraic and geometric dimensions:
. . .However, algebra admits many dimensions (metaphorically speaking). . . . , p. 114
. . .A Plano-planum? But that is, in a nature, huge Monstrum; all Chimera or Centaur most
impossible., p. 137
Wallis, [93], 1693
Pascal approached the “plane-plane” through the concept of integration (see [14]) and also
divided a hypersolid to a sum of smaller parts ( [15]).
It seems, that in the 18th century, it was the right time to add the time (see [5]).
“. . .duration as a fourth dimension”
d’Alembert, [27, pp. 1009–1010], 1754
Lagrange in [57], 1788, used and described n-dimensional Euclidean space without noticing,
while concerning mechanical systems (see [79, p. 348]).
“. . .the problems of mechanics are conceived as being functions of t. Thus we may regard
mechanics as a geometry of four dimensions . . . ”
Lagrange, [58, p. 223], 1797, translation from [15]
There is no motion of a triangle in a plane that could place it to its axial reflection. One could
use a M¨
obius strip instead of a plane, or do the “half-turn” in the space. If one has a simplex in
a three-dimensional space,
“. . .than one needs a system in a space of four-dimensions to make a half rotation. But since
such a space cannot be thought of. . .”
M¨
obius, [65, p. 172], 1827
According to Manning [62, p. 4], this is the first contribution to synthetic geometry of four
dimensions. On the other hand, Jacobi in [49] constructed integral formulas to count volumes
of (n−1)-dimensional spheres, though without any geometric interpretation ( [54, p.76], [26]).
In 1843, Hamilton crossed the bridge to another dimension:
“. . .we should then be able to express the desired product of two lines in space by a
quaternion, of which the constituents have very simple geometric significations. . . ”
Hamilton, [34, p. 47], 1853
Cayley, in 1845, wrote a journal article Chapters in the Analytical Geometry of (n) Dimen-
sions [21], but did not give a multidimensional geometric interpretation (see [26], [54, pp.
76–77]). However, a geometric analogy is an integral part of the Grasmmann’s Lineale Aus-
dehnungslehre, where so-called extensions are generated by changes, and an n-dimensional
affine space is defined ( [26], [54, p.77–78], [79, p. 439-443]).
“. . .and one cannot come to more than three independent directions (changes), while in the
pure extension their number can increase infinitely.
Grassmann, [32, p. XVIII, par. 11], 1844
Further on in his Ausdehnungslehre, 1862, [33] an n-dimensional vector space is defined. It is
a model, which makes a concept understandable ( [26], [54, p. 78], [79, p. 405–406]). So let
Pl¨
ucker take a line in homogenous coordinates (in a space):
3
“. . .As a consequence of the arbitrariness of this choice, the six coordinates, on which the
positions of the two points depend, are reduced to four, which determine the straight line.
Pl¨
ucker, [70, p. 2], 1868
One unique contribution to the multidimensionality remained unnoticed in the 19th century
and was published only 6 years after the author’s death. Schl ¨
afli, in 1852, created an analytic
geometry in n-dimensions with the generalization of Euler’s formula for polyhedra, described
convex regular polytopes by their Schl¨
afli symbols, and derived that there are three of them in an
n-dimensional space for n > 4, and a four-dimensional space is somehow special ( [26], [54, p.
78–81], [79, p. 444–445]):
“. . .Among the 6 characteristics given above as not impossible, two, (3, 3, 3) and (3, 4, 3) are
reciprocal with themselves; the rest consists of two pairs of reciprocal characteristics: (3, 3,
4), (4, 3, 3) and (3, 3, 5), (5, 3, 3). . .. ”
Schl¨
afli, [75, p. 46], 1901 [1852]
Imagine, there is Gauss in front of you, and you want to explain the multidimensionality. If
you were Riemann:
“. . .I have in the first place, therefore set myself the task of constructing the notion of a
multiply extended magnitude out of general notions of magnitude.. . ”
Riemann, [72], 1868 [1854]
Kvasz in [56, pp. 143–150] commented on how Riemann in his works changed the idea of
surrounding ´
a priori given space by forgetting it and constructed geometric objects implicitly.
After the foundation of the multidimensionality, complex and non-euclidean geometry in
the 19th century, the time has changed (relatively speaking), too (see [79, 517–520]).
Furthermore, I will operate on the use of complex quantities in a manner not yet customary in
physical investigations, namely, instead of t, with the compound it...”
Minkowski, [63], 1908
One can think of mathematics as a science that gives unshakable answers, but in mathematics,
answers raise new questions. At the beginning of the 20th century, fractal geometry was born
(see also [56, p. 62], [79, pp. 512–514]).
“. . .we present an explanation of the p-dimensional measure that extends to non-integer
values of pand shows fractal dimensions as possible. . .”
Hausdorff, [40], 1918
After the foundation of relativity, the physical world of geometric space and time has been
united in four dimensions and axiomatically rooted by Weyl ( [79, pp. 443, 521]). See his
axiom of dimension:
“There are nlinearly independent vectors, but each (n+ 1) are linearly dependent.”
Weyl, [97, p. 17], 1919
We tried to show in brief, how the mutlidimensionality become an integral part of mathematical
thinking. However, we should not forget, that this path was full of denial, and in the following
sections, we explain some methods of approaching the fourth dimension.
4
2.2 Visual dimension
There is only one geometry, but there are different methods of examination of geometric ob-
jects. It is by analytic deduction how we prove and verify hypotheses. It is by synthetic impli-
cation how we obtain an intuition and create hypotheses. In geometry, constructions of figures,
visualizations or models of geometric objects occupy a special position. But how to grasp four
and more-dimensional space? Let us think of a three-dimensional analogy first: if we draw a
cube on a paper, we usually draw its projection (either parallel or central). But would a two
dimensional creature understand our picture as a three dimensional cube or as a bunch of seg-
ments instead? We can try several graphical methods of explanation. Since the creature should
understand planar geometric properties, we can draw consecutive planar sections of the cube.
We might create the net of faces of the cube. Or, to grasp the three-dimensionality of the cube,
we often draw its shades and shadows. A certain spatiality is also seen by a motion of the cube,
and hence we could draw projections of consecutive locations by this motion, or use animations
and interactive tools. At last, it is much easier to illustrate a particular object such as a cube,
which is a simple generalization of a square (assuming the creature has a good knowledge of
a square), than to visualize an n-dimensional space itself. More precisely, imagine a plane
. . . . Do we really imagine a plane in its whole infiniteness, or do we rather imagine its small
(rectangular, triangular, or pancake-like) piece? As if we only need some local generator of the
plane and abandon the idea of the space itself, similarly to Grassmann and Riemann.
It was Monge [66] and his followers in the beginning of the 19th century, who established
descriptive geometry as a rigorous constructive method of investigation of geometric objects
and as a mathematical theory itself. The subject of descriptive geometry is a projection of a
spatial object into a plane. However, the sense of descriptive geometry is not in the mere tech-
nical drawing; with descriptive geometry one solves how to precisely, point-by-point, construct
objects from the given conditions, how to use geometric transformations such as affinities and
collineations, how to use special properties of shapes, how to find reasonable positions for visi-
bility and visualization, how to use projective, affine or metric properties, and many more. De-
scriptive geometry is also a playground, where one can build new objects or observe properties
of known objects without necessity of algebraic representation. For general history of descrip-
tive geometry see the recent book [11]. Graphical interpretations of four dimensional geometry
started to appear around 1880’s [92, pp. 107–121]. Scheffler in [74, pp. 139–159] described and
illustrated generating tetrahedral sections of polytopes. Especially regular polytopes became
a fruitful area for visual demonstrations. Visualizations of boundaries of regular polytopes in
symmetric arrangements in three-dimensional perspective and also planar diagrams were given
by Stringham [85]. Schlegel [76] created very clear central projections of regular polytopes,
nowadays called “Schlegel diagrams”. Schlegel’s three-dimensional models are to be found
in G¨
ottingen Collection of Mathematical Models and Instruments [1] (for details on “four-
dimensional models” see [91]). A construction of a projection of a hypercube on a hypersphere
by Goursat is given in [31]. Figures of properties in four-dimensional space have become in-
tegral part of textbooks, e.g. the second part of Veronese’s textbook on geometry [89], which
is devoted to four and n-dimensional spaces, contains many explanatory illustrations. The net
of a hypercube, consisting of eight cubes was depicted by Puchta in 1884 [71], [92, pp. 118,
119]. A comprehensive treatise on four and multidimensional polytopes in different projections
was given by Shoute in 1902 [78]. Shoute also lectured Boole Stott, who described and de-
picted derivation of semi-regular polytopes in [13]. Several contributions on four-dimensional
5
descriptive geometry using projections into plane appeared in the beginning of the 20th cen-
tury. Four-dimensional descriptive geometry is explained in the comprehensive book [51] from
1906 by Jouffret. He described projections of the four-dimensional space onto a plane and
constructed several complicated examples of polytopes and their sections. Apart from the ax-
onometric projection, Jouffret used orthogonal projections similar to Monge’s projection with
reference planes in plane quadrants A(x1, x3), B(x3, x2), C (x2, x4), D(x4, x1)merged in the
drawing plane such that positive x1and x3values are in the quadrant A, and positive values of
on x2and x4are in the quadrant C. Therefore, each point has four conjugated images. Similar
idea was described by Eckhart in 1926 [29, pp. 18–37]. A complete book on four dimensional
descriptive geometry with orthogonal projections of elementary objects into a drawing plane
is given by Lindgren and Slaby [60]. Stachel in [82] gave a proof of the four-dimensional
right-angle theorem in the orthogonal projection onto a plane. Axonometric projections of hy-
perspheres into planes are described by Weiss in [94]. Casas in [20] used his unusual method of
polar perspective — a system of flattened perspective projections onto spheres to visualize the
fourth dimension. Visualizations of shadows of polytopes are explained by Chilton in [22,23].
Recently, an analogy to Monge’s projection onto multiple planes was described in [30].
Modern technology has brought new perspectives on geometric visualization. With com-
puter graphics, we can create figures impossible to draw by hand, or even more, animations of
such objects. But, instead of synthetic constructions, analytic representations of objects are usu-
ally more useful for computers. Techniques of visualization of projections of four-dimensional
surfaces are to be found in [47, 83, 100, 110]. Banchoff wrote a complete textbook [9] on
four-dimensional visualization. Motion of a die is incorporated in [64]. Consecutive sections
of pentachoron with hyperplanes placed on a parabolic curve are in [52]. Shadows and light-
ing of four-dimensional surfaces are studied in [37]. Other possibilities were brought to life
with 3D printing. Constructions of three-dimensional physical models of projections of poly-
topes are described in [81]. Segerman in [80] created an exquisite collection of models of
three-dimensional central and stereographic projections (also [77]) of more complicated four-
dimensional objects. Moreover, Segerman and Hart in [39] constructed a 3D printed projection
of a hypercube with a symmetric monkey in each cell such that connections of its hands, legs,
and head compose the symmetry of the quaternion group. This example also represents the
value of mathematical visualization not only to visualize some object, but to use it as a tool
of investigation of some four-dimensional phenomenon without an analogy in lower dimen-
sions. Hanson accented the visual approach to quaternions in his textbook [35]. A different
application is in [99] to visualize scalar functions. Nontrivial four-dimensional objects arise
in topology. One particular example is the Hopf fibration — a mapping from a 3-sphere to a
2-sphere. Animations, visualizations of the Hopf fibration and stereographic images are given
in [4, 8, 24, 50, 53, 55]. Another direction into computer graphics was set by incorporating
interactivity. Interactive methods have given us new opportunities to examine geometric rela-
tions and create hypotheses. Considering four-dimensional geometry, interactive methods were
presented in [10, 12, 25, 36, 38, 44, 87, 109].
With more and more complicated geometric objects, the laboriousness of precise hand-
drafted constructions, created by methods of classical descriptive geometry, became unsustain-
able. On the other hand, computer visualizations based on analytic representations lack the
synthetic creation. Recently, we tried to merge interactive methods of computer graphics and
methods of descriptive geometry in [102–104,106,107]. An introduction into this method will
6
be given in Section 3.
2.3 Expository dimension
In the previous sections we have seen that understanding and visualization of the fourth di-
mension enriched each other. Yet still, our spatial ability does not allow us to imagine four-
dimensional objects.
The most common method of investigation is to use a heuristic analogy (for details on
the use of analogy in education see [6]), popularized in four dimensions by Hinton in 1880
(see [46]). Let us have a point and move it in one direction to a segment; move the segment
in the second, perpendicular to the first, direction to a square; move the square in the third,
perpendicular to all the previous, direction to a cube; move the cube in the third, perpendicular
to all the previous, direction to a hypercube; and so on. Now, what we can try to imagine is an
extension of the dimension; but let us try it backwards, starting with a cube. Look at the front
face, so we may say that the cube is a figure behind the front square, look at a side of a square,
and so the square is a figure behind the segment, and look at the segment from its direction,
and the segment is a figure behind the point. At this point, you can try to explain what is a cube
to the befriended two dimensional creature. Instead of extending the space, we may think that
we only see the edge view of some object and the rest is behind, or maybe deeper inside, the
square.
If the analogy is straightforward, it is a powerful weapon for solving dimensional problems.
But let us try another example: let us have a circle and its inner point in a plane. It is impossible
to move the point out of the circle in the plane. Therefore, we move the point in the third
dimension and jump above the circle. What is the analogy in the three-dimensional space? Is
it a point in a sphere, a circle around a line, two interlinked circles? Actually, whatever it is,
the solution would be the same, we only need to jump through the fourth dimension. We may
have shaken the concept of analogy a little, hence the last example: conics are sections of a
conic surface, and so the regular conics are collinear images of a circle, and collinear images
preserve collinearity. Analogically, when we cut a hypercone with a 3-space, three-dimensional
quadrics should be collinear images of a sphere. But are they? What about ruled quadrics with
generating lines. Due to collineation, if there is a line on the image, there must be a line on
the original quadric. In addition to a hypercone with a spherical base (for unruled regular
quadrics), we need another hypercone with a one-sheet hyperboloidal base (for ruled regular
quadrics). Here, the generalization was non-trivial, and the problem should be solved directly in
the four-dimensional space (both analytically and synthetically). On the other hand, planimetric
problems are sometimes solved through a three-dimensional explanation. Recall Desargues’s
theorem with two triangles ABC and A0B0C0such that AA0, BB0, C C0are concurrent, then
intersections of AB, A0B0;B C, B0C0;CD, C0D0are collinear. The property seems obvious
when we look at it as a triangular base and a section of a pyramid. This way, we can use the
hyperdimensionality to solve problems in the lower space (see [48, 102], [105, pp. 37–67]).
And did we not do such thing with defining conics and quadrics through more-dimensional
objects? After all, models of projective and hyperbolic 3-spaces, 3-manifolds, complex plane,
etc., could be conveniently examined in an embedding real four-dimensional space.
Each-dimensional space brings its own problems and a proof, or a construction, is not
always a “trivial generalization”. We can explore it through valid analogies, but at least the
7
same effort should be made to find new objects and their properties directly. And in some cases
it may even be prosperous for lower dimensions.
2.4 Narrative dimension
Analogy and paradox! We have discussed that the unimaginable geometry of the fourth dimen-
sion is related to geometry of the three-dimensional space by generalization, and that it also
carries new properties, not uniquely restrictable, to lower dimensions. Through the doors of
analogy and paradox, dimensionality become fertile soil for metaphors or stories. Such narra-
tives are to be found in philosophy, religion, mysticism, literature, painting, architecture, and
many more (see e.g. [43,45,59, 95]). Instead of discussing meanings of interpretations, we give
a few examples of discussed problems that may enrich the geometrical point of view. Other-
wise, why would we wonder about the fourth dimension if there were no problems to solve.
One way to visualize the dimensionality of an object is through its projections or shadows.
Such a projection is used by Plato in his well-known Allegory of the cave (Plato, [69, p.224],
4th century BC):
“. . .the truth would be to them just nothing but the shadows of the images”
And later on, what would happen if one is taken to the “higher dimension”:
“. . .He will require to get accustomed to the sight of the upper world. And first he will see the
shadows best, next the reflection of men and other objects in the water, and then the objects
themselves; next he will gaze upon the light of the moon and the stars; and he will see the sky
and the stars by night, better then the sun, or the light of the sun, by day?”
Hinton in his essay What is the fourth dimension [46] introduced moving figures as traces of a
system of threads with consciousness, and he asked:
“. . .Is it possible to suppose that the movements and changes of material objects are the
intersections with a three-dimensional space of a four-dimensional existence? Can our
consciousness be supposed to deal with a spatial profile of some higher actuality?”
Further on, Hinton discussed the relation of four and three-dimensional beings through the
analogy of three and two-dimensional beings. This analogy from lower dimension was stated
in 1870 by Helmholtz (translation from [42]).
“Let us, as we logically may, suppose reasoning beings of only two dimensions to live and
move on the surface of some solid body. We will assume that they have not the power of
perceiving anything outside this surface, but that upon it they have perceptions similar to
ours.”
The concept was also used in the famous popularizing book Flatland1by Abbott [3]. He de-
scribed a flat world (and even less dimensional worlds) with its geometric properties. Sections
1At this place, we should explain the title of the article. The author himself is an example of how a story may
unconsciously imply research in geometry. About two months after reading Flatland, while preparing a lecture on
multidimensional analytic geometry, a question raised: “Why should we do this analytically?” The obvious answer:
“Because we cannot explore the fourth dimension with synthetic geometry!” was rejected immediately with the concept
of the double orthogonal projection described in Section 3.
8
16 and 17 discuss long-lasting series of mathematical arguments when the three-dimensional
Sphere tries to (unsuccessfully) persuade the main character — Square, that there are three di-
mensions. After failing in this task, the Sphere abducts the Square into the third dimension and
here is what the Sphere looks like:
“Prostrating myself mentally before my Guide, I cried, ”How is it, O divine ideal of
consummate loveliness and wisdom, that I see thy inside, and yet cannot discern thy heart, thy
lungs, thy arteries, thy liver?” ”
The opposite effect, when a higher dimensional being sees the inside of a two dimensional
being, is masterfully depicted in the episode 14, series 7, of the animated TV show Futurama.
When a group of characters enters the two-dimensional world, the Professor explains several
questions. For example, when Leela has troubles swallowing an apple:
“I guess it’s because our two ended digestive system can’t exist in this dimension. I suppose
that could be an issue.”
Approaching problems of the three-dimensional world, how did Alice get through the look-
ing glass [18]? Is there not implicitly stated M ¨
obius’s problem about motion of a simplex to
its own reflection? A special position in four-dimensional phenomena obtained in our three-
dimensional world belongs to an astrophysicist and spiritualist Z¨
ollner, who performed series
of controlled experiments (e.g. untying knots) on the existence of the four-dimensional space
(see [111]). In the early 20th century, a cubist Duchamp studied four-dimensional geometry
from Jouffret and other authors and experimented with his paintings using sections, shadows,
and optical illusions. Another cubist, Max Weber, pictured sequences of spatial sections in his
Interior of the Fourth Dimension in 1913. The geometry of a hypercube creates the plot of a
story And He Built a Crooked House [41] in a pulp magazine, where an architect Quintus Teal
tries to persuade his client to buy a four-dimensional house.
“Bailey studied the wobbly framework further. “Look here,” he said at last, “why don’t you
forget about folding this thing up through a fourth dimension—you can’t anyway—and build a
house like this?”
“What do you mean, I can’t? It’s a simple mathematical problem—”
“Take it easy, son. It may be simple in mathematics, but you could never get your plans
approved for construction. There isn’t any fourth dimension; forget it. But this kind of a
house—it might have some advantages.” ”
After a while, Teal is successful at least in building a three-dimensional net of the hypercu-
bical house. But there is an earthquake, and the house folds into its four-dimensional form.
The characters are trapped in the house exploring its inner geometry. For example, they see
themselves four rooms away. The story inspired an architect Capanna into a construction of the
architectural design of the crooked house in [16]. The net of a hypercube is also used in the
famous Dali’s painting Corpus Hypercubus (see mathematician Thomas Banchoff’s lecture for
a detailed description [7]). Similar idea with a pentachoron (5-cell) house is used in the story
about Pento and his creditor Bill on the website [2] devoted to the four-dimensional geometry.
“Out Bill ran, and never came back; “For,” he swore, “the kitchen was behind this door, it was
one room of the connected four. This place must be haunted, and its owner unwanted; for how
could 5 rooms be interconnected and fully surrounded, each with 3 doors and a trapdoor?””
9
After all, an architect and futurist Fuller presented his “four-dimensional” tower and Dymaxion
house in 1920’s. If we accept Peterson’s interpretation [68], the summit of imagination of inner
structure of four-dimensional objects, particularly a 3-sphere, is surprisingly reached by Dante
in his Paradiso in Divine Comedy. In a certain point, it seems like Dante was describing his
universe from the equatorial 2-sphere of a 3-sphere.
Finally, we cannot omit the time dimension. See what Wells developed in his novel The
Time Machine [96] from 1895:
“There are really four dimensions, three which we call the three planes of Space, and a fourth,
Time. There is, however, a tendency to draw an unreal distinction between the former three
dimensions and the latter, because it happens that our consciousness moves intermittently in
one direction along the latter from the beginning to the end of our lives.”
A more sophisticated time machine TARDIS (Time And Relative Dimension In Space) plays
a fundamental role in the TV series Dr Who. Several episodes deal with explanations of para-
doxes based on dimensionality, for instance in the episode The Space Museum, 1965, the Doctor
says:
“. . .You know I don’t mind admitting I’ve always found it extremely difficult to solve the fourth
dimension. And there we are, face to face with the fourth dimension. You know, I think the
TARDIS jumped the time track and ended up here, in this fourth dimension, hmm,
extraordinary.. . . ”
The fourth dimension does not only cause problems, it can also solve some of them. For
example, how to jump around a black hole in Interstellar:
“Rom (taking paper): In the illustrations, they’re trying to show you how it works. So they
say, you want to go from here to there. But it’s too far, right? So a wormhole bends space like
this (bending the paper, so the two points coincide), so you can take a shortcut through a
higher dimension. To show that they’ve turned three-dimensional space into two dimensions
which turns a wormhole into two dimensions, a circle. What’s a circle in three dimensions?”
We have shown several metaphors and stories, through which one can state new problems
and discover properties of the four-dimensional space. Furthermore, solving real or even unreal
problems often deepens our understanding of the geometric concept.
3 Double orthogonal projection of the 4-space
In this section, we give several examples to manifest how to get deeper understanding of the
fourth dimension through visualizations. We use the method of double orthogonal projection
of the 4-space, which is an analogy to Monge’s projection of the 3-space (see [103,104]).
In Monge’s projection, a point P(xP, yP, zP)is orthogonally projected into two mutually
perpendicular planes π(x, y)(horizontal) and ν(x, z)(vertical). One of these planes, say ν, is
chosen to be a drawing (picture) plane, and πis rotated about the x-axis to this drawing plane
such that zand yhave opposite directions. Therefore, the orthogonal images — the top view
P1(xP, yP)in π(after the rotation) and the front view P2(xP, zp)in νlie in the drawing plane.
The directions of the orthogonal projections are both perpendicular to the x-axis, hence in the
10
drawing plane, P1and P2lie on a perpendicular to the x-axis called the ordinal line (line of
recall), e.g. the line created by the overlapping images of the projecting rays.
Analogically, in the double orthogonal projection of the 4-space onto two mutually per-
pendicular 3-spaces, a point P(xP, yP, zP, wP)is orthogonally projected into two mutually
perpendicular 3-spaces Ξ(x, y, z)and Ω(x, y, w). We choose the 3-space Ωto be a modeling
3-space and rotate Ξabout the plane π(x, y)to the modeling 3-space. Well, the reader could
have noticed that the previous step had been unimaginable, but think of the poor two dimen-
sional creature, who cannot imagine even the rotation of the planes about their intersecting line
in the Monge’s projection, and yet we can draw it in a plane. After the rotation, the point Phas
two orthogonal images P3(xP, yP, zP)in Ξand P4(xP, yP, wP)in Ω. Both P3and P4have
common xand ycoordinates, and so they lie on a perpendicular to π(x, y), called the ordinal
line.
While in Monge’s projection we can see everything on the drawing plane in a single view, in
the generalization, we should think that we can approach any location in the modeling 3-space.
Certainly, our figures are on a two-dimensional paper or screen, but the three-dimensional
visibility does not play any role at all. The same would happen, if you draw a picture, be it
a square, on a paper for the two-dimensional creature. It will see at most two edges, but for
us the square is fully visible. Therefore, the essence of interactivity is very important in our
approach, and we attach supplementary GeoGebra book [101], where the user is able to move
inside the modeling 3-space or manipulate with certain elements.
3.1 Points, lines, planes, and 3-spaces
Figure 1: Double orthogonal projection of a point, line, and plane.
https://www.geogebra.org/m/asyp57w6#material/sw4ysvpu
In the double orthogonal projection, a point, a line, and also a plane (in contrast to Monge’s
projection) have two conjugated images in the modeling 3-space (Figure 1). In general, images
11
of a 3-space would cover the whole modeling 3-space. Thus, we only draw planar intersections
of the 3-space with the reference 3-spaces Ξand Ω, called traces (Figure 2). Planes of a 3-space
parallel to its traces (an analogy to frontal and horizontal lines) may be conveniently used to
locate a point in this 3-space. Images of a line perpendicular to a 3-space are perpendicular to
its traces.
Figure 2: Trace planes ξand ωof a 3-space, its point P, and a perpendicular pthrough Pto the
3-space.
https://www.geogebra.org/m/asyp57w6#material/g6gvqfnp
In the following example we give a synthetic alternative to the ordinary analytic solution.
Let us have two distinct planes ρand σin the 4-space. What are the possibilities of their com-
mon positions? Analytically in R4,ρis given by a point Rand linearly independent nonzero
vectors u1, u2;Σis given by a point S6=Rand linearly independent nonzero vectors v1, v2.
The dimension of the subspace spanned by the vectors u1, u2, v1, v2can be 2, 3, or 4. In case
of 2, the planes are identical (if the vector (R−S)is linearly dependent with the rest) or totally
parallel (if not). In case of 3, the planes have one common direction, and they intersect in a line
(if the vector (R−S)is linearly dependent with the rest), or they are parallel in one direction
(if not). In case of 4, they intersect in a point. Instead of using linear algebra, we could explic-
itly draw respective sketches (Figure 3). Let us have a fixed plane (black) and choose different
positions of the images of the second plane (red). This way, we can decompose a solution in
the 4-space into solving positions of the images in the 3-spaces Ξand Ω(depicted in general
positions of planes with respect to the reference system). Planes are identical when both images
of both planes are identical, parallel if parallel (Figure 3, top left). For all the other cases, let
the images in Ωintersect in a line (blue). Through the intersecting line we construct its ordinal
plane (green), which also intersects the second image of the fixed plane in a line (blue). If the
image of the red plane in Ξis:
1. parallel (and not incident) to the Ξ-image of the blue intersecting line, the position of the
planes is parallel in one direction (Figure 3, top right) (similarly for parallel images in Ωand
12
Figure 3: Positions of two planes.
https://www.geogebra.org/m/asyp57w6#material/cvnbfafr
intersecting images in Ξ);
2. passing through the Ξ-image of the blue intersecting line, the position of the planes is inter-
secting in one line (Figure 3, bottom left);
3. intersecting the Ξ-image of the blue intersecting line in a point, the position of the planes is
intersecting in one point (Figure 3, bottom right).
There are no other positions of planes in the 3-space Ω, and a plane and line in the 3-space Ξ.
3.2 Hypercube
Constructions of Monge’s projection of a section of a cube with a plane and a shadow that
the cube casts on a plane are elementary exercises in descriptive geometry. Let us choose the
plane to be the bisecting plane of a space diagonal of the cube. For simplicity, the cube is
in a special position with the top and front images as squares. The section of the cube is a
regular hexagon, which is observed in its true shape after the rotation of the section plane to the
drawing plane (Figure 4, left). Furthermore, let us have a directional lighting in the direction
of the space diagonal. Thus, the shadow, which the cube casts on the bisecting plane, is the
image of the cube in the orthogonal projection. Again, the image is a regular hexagon, visible
13
in its true shape after the rotation into the drawing plane (Figure 5, left). And so what is the
four-dimensional analogy?
(a) Section of a cube with the bisecting plane and
its true shape.
(b) Section of a hypercube with the bisecting 3-
space and its true shape.
Figure 4: https://www.geogebra.org/m/asyp57w6#material/afr3xnrb
(a) Projection of a cube into the bisecting plane
and its true shape.
(b) Projection of a hypercube into the bisecting
3-space and its true shape.
Figure 5: https://www.geogebra.org/m/asyp57w6#material/afr3xnrb
Let us have a hypercube with a hyperspace diagonal and its bisecting 3-space (Figure 4,
right). The hypercube is placed in the reference system such that its conjugated images are
14
cubes. The traces of a bisecting 3-space through the center of the cube are easily constructed
with the use of parallel planes and perpendiculars (see Figure 2). The spatial section of the
hypercube with the bisecting 3-space is an octahedron. Its images are affinely distorted, but
after the rotation of the bisecting 3-space about its trace onto the modeling 3-space (which
is another spatial affinity in the modeling 3-space), we obtain the true shape of the section —
regular octahedron, a Platonic solid. Independently on the visualization, the idea of the analogy
to the cube would be to observe that the planar section of the cube regularly crosses each of its
six faces, while the spatial section of the hypercube crosses each of its 8 cells.
The construction of the projection of the hypercube into the bisecting 3-space is carried out
in the similar manner (Figure 5, right). The intersections of the light rays through the vertices
of the hypercube and the bisecting 3-space are vertices of a dodecahedron in the bisecting 3-
space. Again, its images are affinely distorted, but after the rotation into the modeling 3-space,
the true shape of the projection is a Catalan solid — rhombic dodecahedron. In contrast to
the section, we obtained a different solid dimensionally analogous to a regular hexagon. The
idea in this case could be: when we observe a cube in the direction of its space diagonal, with
a vertex in front of us, we see three parallelogram faces (distorted squares) connected in a
regular hexagon; in the four-dimensional case, when we observe a hypercube in the direction
of its hyperspace diagonal, we see four parallelepipeds (distorted cubes) connected into one
rhombic dodecahedron.
One further unavoidable remark, in three dimensions, the hexagonal section of the cube
is inscribed in the hexagonal image in the orthogonal projection such that the vertices of the
section are the centers of the sides of the image. In four dimensions, the regular octahedron is
inscribed in the rhombic dodecahedron such that the twelve edges of the octahedron are longer
diagonals of the twelve rhombic faces of the dodecahedron.
3.3 3-Sphere and a sterographic projection
If we were Cubes, how would a 3-Sphere from the higher space argue to persuade us into
the existence of the four-dimensional space? One way of introduction is to show itself in
our 3-space. Passing through our 3-space, we would see its sections as a point extending
to a still bigger 2-sphere, reaching the equatorial 2-sphere, shrinking into a point again, and
disappearing. In the double orthogonal projection, these were sections of a 3-sphere with the
planes parallel to one of the reference 3-spaces (e.g. Ξin Figure 6, left). Intersections of a
3-sphere with 3-spaces, planes, and lines in general positions are described in [106].
Assume we live on the Earth (be it a sphere). One of the fundamental problems is to create
its planar map. One of the solutions is to construct a stereographic projection, for example,
from the North pole to the equatorial plane. Lengths would be distorted, but angles would be
preserved. On the other hand, from the image (map), we are able to study the original object
(the Earth). In the same manner, to study the geometry of the 3-sphere, we can construct its
stereographic projection from a pole Nto the equatorial 3-space (Figure 6, right).
Let us have a hyperspherical tetrahedron ABCD in a special position on a 3-sphere such
that the face ABC is on the equatorial 2-sphere (its image is the 2-sphere in the 3-space Ω), the
face ABD is on the great 2-sphere in the 3-space perpendicular to the z-axis (its image is the
2-sphere in the 3-space Ξ), and the face ACD is on the 3-sphere in the 3-space perpendicular to
the x-axis (hence the images of the 3-sphere are conjugated circles with their inner points, such
15
(a) Parallel sections of a 3-
sphere. (b) Stereographic projection of a hyperspherical tetrahedron ABC D.
Figure 6: Left https://www.geogebra.org/m/asyp57w6#material/btpju9qc
Right https://www.geogebra.org/m/asyp57w6#material/uuty7bh5
as the prime meridian in the Monge’s projection). In this position, the stereographic images
AS, BSand DSof the vertices A, B and Dare incident with their Ξ-images A3, B3and D3.
The stereographic image CSof the point Cis constructed as the intersection of the line NC
and the equatorial 3-space. The true shapes of the edges of the tetrahedron are circular arcs
affinely distorted in the images to elliptical arcs (or lines, when passing through N). However,
due to the conformity of the stereographic projection, the stereographic images of the edges are
again circular arcs (or lines).
3.4 Hypercones
If the presence of the 3-Sphere was not satisfactory, it could bring some singular friends. We
have already stated the problem with analogy of planar conic sections of a circular cone (more
properly, we should talk about a conical surface, but we only visualize its parts) in Section 2.3.
If the section plane does not contain the vertex of the cone, each conic is a collinear image of
the base circle. A collineation perserves collinearity, i.e. lines map to lines. Therefore, the
four-dimensional analogy of a hypercone with a spherical base is not enough if we consider
ruled quadrics. While unruled ellipsoids, two-sheeted hyperboloids, and elliptic paraboloids
16
(a) Parallel sections of a spherical hypercone. (b) Parallel sections of a one-sheet hyperboloidal
hypercone.
Figure 7: Left https://www.geogebra.org/m/asyp57w6#material/wbtxzyzv
Right https://www.geogebra.org/m/asyp57w6#material/e6zwwfj6
could be considered as collinear images of the base 2-sphere; ruled quadrics: one-sheet hy-
perboloids and hyperbolic paraboloids are collinear images of a base one-sheet hyperboloid.
Let us construct consecutive sections of a spherical and one-sheet hyperboloidal cone parallel
with the reference 3-space Ξ. The sections of the spherical cone start with the base 2-sphere,
shrinking into a point — the vertex, and again extending into bigger 2-spheres (Figure 7, left).
The sections of the one-sheet hyperboloidal hypercone start with the base one-sheet hyper-
boloid, becoming narrower hyperboloids, shrinking into a circular cone (with the same vertex
as the hypercone), and growing into one-sheet hyperboloids again (Figure 7, right). Quadratic
sections of hypercones with 3-spaces in general position are described in detail in [107].
3.5 Complex number plane
A cartesian plane R2with real and imaginary axes (also Gaussian, or Argand plane) is often
used for visualizations of complex numbers, their properties, and representations. The set of all
complex numbers creates the complex number line C. To visualize the complex number plane
C2, the four-dimensional real space R4is necessary.
Let us have a point Pin C2. The coordinates of Pare P[aP, bP]for aP, bP∈C. Later on,
aPhas real and imaginary parts aP[xP, yP]and bPhas parts bP[zP, wP], for xP, yP, zP, wP∈
R, and so the point P∈C2is represented by four real coordinates P[xP, yP, zP, wP]. Further-
more, the complex numbers aPand bPhave trigonometric representations aP=ra(cos α+
isin α)and bP=rb(cos β+isin β)for α, β ∈R(but for the sake of visualizations, we only
need the interval h0,2π)) and ra=qx2
P+y2
P, rb=qz2
P+w2
P.
17
Figure 8: Visualization of a point in C2.
https://www.geogebra.org/m/asyp57w6#material/jkunbar4
To depict the point P, we swap the axes yand zin the reference system of the double
orthogonal projection (Figure 8). This way, the real parts of aPand bPare in the plane π(x, z),
and the reference 3-spaces have coordinates Ξ(x, z, y)(upper) and Ω(x, z, w)(lower). The
point Pwill have two images P1(xP, zP, yP)and P2(xP, zP, wP)in the modeling 3-space
(Figure 8). Moreover, we can construct the rotated point Prin the modeling space to obtain the
modulus |P|, i.e. its distance from the origin. Certainly, the point Plies on a 3-sphere with the
center in the origin and the radius |P|=qr2
a+r2
b=qx2
P+y2
P+z2
P+w2
P.
Even though it is essential to visualize algebraic structures for explanations and intuitive
reasoning, it is not our aim to be more complex in this field, within this paper. From this point,
one can investigate relations between visual and algebraic properties of C2.
3.6 Torus and the Hopf fibration
We have described, through analogies with lower dimensional situations, the visualization of
a 3-sphere and its stereographic projection in Section 3.3, as well as the visualization of C2
in Section 3.5. Putting it all together, we obtain a surprising result and its visualization. The
Hopf fibration is a mapping between a 3-sphere and a 2-sphere such that circles on the 3-sphere
map to points on the 2-sphere. More precisely, a unique point on the 2-sphere corresponds to
a unique circle on the 3-sphere. Even more, two distinct points on the 2-sphere correspond to
two interlinked circles on the 3-sphere. Consequently, points on a circle on the 2-sphere map to
18
circles on a torus on the 3-sphere. To see circles properly, we conveniently use a stereographic
projection. Our goal is to show visualization of this phenomenon, which has no analogy in
lower dimensions. For a reader interested in technical details, we recommend approachable
explanations [61,88] with graphical demonstrations.
Figure 9: Visualization of the Hopf fibration.
https://www.geogebra.org/m/asyp57w6#material/pzahq4uc
In the double orthogonal projection, the conjugated images of the 3-sphere are 2-spheres in
the 3-spaces Ξand Ω. Keeping the change of coordinate axes from Section 3.5, the 2-sphere
in the Ξ-image (upper) is the section of the 3-sphere with a 3-space parallel to Ξthrough the
center of the 3-sphere. The Ξ-image is also used to be the 2-sphere from the Hopf fibration.
Now, if we choose a point P[aP, bP]∈C2on the 3-sphere, the Hopf fibration is the following
function between the spheres C2(3-sphere) →C×R(2-sphere):
f(aP, bP) = (2aPbP,|aP|2− |bP|2).
Moreover, for λ∈Csuch that |λ|2= 1, we have f(aP, bP) = f(λaP, λbP). That means, not
only the point P, but also all the points of the circle generated by λon the 3-sphere through
Pmap to the same image on the 2-sphere. In [108] we have proposed a reversed synthetic
construction of the circle on the 3-sphere from the given point on the 2-sphere. A point on the
2-sphere, images of its corresponding circle (Hopf fiber) on the 3-sphere, and its stereographic
projection in the 3-space Ωare visualized in Figure 9. Additionally, in the figure, a circle on
the 2-sphere, images of its corresponding Hopf torus on the 3-sphere (cylinders) and the torus
in the stereographic projection are constructed.
4 Conclusion
Paradox and analogy! To the Hopf fibration from Flatland. Vice-versa, we showed a visualiza-
tion of a four-dimensional phenomenon, which is not a generalization of the lower dimensional
19
case. Therefore, the construction had to be carried out directly in the four-dimensional space.
On the contrary, instances of this construction were based on analogies. Our method of vi-
sualization itself is an analogy of Monge’s projection. We tried to demonstrate this method
as a convenient and powerful form of representation of four-dimensional objects, and as a
playground in which we can construct and explore new objects. Throughout the survey of
visualizations, we always crossed the border between an analogy from the three-dimensional
space and newly obtained properties in the four-dimensional space. Thus, the four-dimensional
geometry is not just a trivial generalization. It creates fascinating problems intersecting with
our physical world, and it places worthwhile metaphors investigated not only in mathematics
and physics, but also in philosophy and art.
At last, we have touched several areas: be it hyperquadrics, complex geometry, stereo-
graphic projection, manifolds embedded in a 3-sphere; each of them extendable for further
research. However we have not touched many areas waiting for a proper investigation: be it
hyperbolic geometry, space-time, hypersurfaces of higher degree, geometric transformations;
and methods of visualization such as the linear perspective and axonometries; and different
kinds of presentation with 3D models, virtual, or augmented reality.
5 Acknowledgment
The work was partly supported by the grant No. CZ.02.3.68/0.0/0.0/16 038/0006965, Faculty
of Education, Charles University.
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