characteristics in the fourth dimension
TRANSCRIPT
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Characteristics in the Fourth Dimension
Angela C. Wood
NSF Scholar 2002-2003
Virginia Commonwealth University
Submitted: August 2004
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Characteristics in the Fourth Dimensions
Abstract
This paper examines the basic characteristics of the fourth dimension, its
history, as well as characteristics of four-dimensional figures. Using properties of
figures in one, two and three dimensions we are able to determine the properties
of four-dimensional figures. Using simple figures, such as circles and squares
makes the fourth dimension more tangible for those students who have never
been exposed to it.
What is the fourth dimension?
Some believe that the fourth dimension represents time and others believe
that it represents a direction. Let us focus on the idea that the fourth dimension
is a direction perpendicular to all directions in normalspace, as it is easiest to
visualize. Examining patterns helps to understand this phenomenon. For
example, an object in the first dimension consists of only one of the fundamental
units, such as a line having only length. An object in the second dimension
consists of two of the fundamental units, such as a square having length and
width. It follows that an object in the third dimensions consists of three of the
fundamental units, such as a cube having length, width and height. Although
difficult to imagine, an object in the fourth dimension consists of four units, length,
width, height and the unknownunit projecting in the fourth dimension. To help
visualize this unknownunit, look at the corner of a room. Notice all of the
intersecting lines and imagine a fourth intersecting line perpendicular to the other
three.
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This is a difficult concept to grasp. It also helps to imagine creatures living
in zero, one and two dimensions. If a bird were trapped at one particular point in
space, it could not move in any direction. Therefore is has zero degrees of
freedom. If a bird were trapped in a small tube just wide enough for it to travel
backwards and forwards it has one degree of freedom. If a bird were trapped on
a two dimensional plane and was unable to fly, it would posses two degrees of
freedom; meaning it could only travel backwards, forwards, left and right.
Obviously a bird trapped in a three dimensional space could travel in what we
think of as our world. Now imagine that the bird is only capable of seeing
forward. Would the bird trapped in zero dimensions be able to see what was
behind it? Would the bird trapped in the tube be able to see what was to the left
and right of it? Would the bird trapped in the two-dimensional plane be-able to
see what was above and below it? The answer to the following questions is no.
See diagrams below:
Zero One
dfdf
Two Three
df df
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Therefore, trapped in a three-dimensional world, we are unable to see in this
unknown fourth dimension perpendicular to all of the other directions. [10]
Edwin A. Abbot wrote a book entitled Flatland in 1884 that describes the
phenomenon just discussed. Flatland consists of a world of two-dimensional
creatures. The towns people are triangles, squares, pentagons, etcand as the
number of sides increased the higher the social status of the people in society.
The circle represented the most prestigious being of all. In Flatland, the people
can only see lines and points; they cannot see height. The following diagram
gives Flatland from our perspective as well as from theirs.
View from above.
View from Flatland.
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The society lived peacefully until one day a creature from Space-land visited.
This creatures size varied continuously as it moved. The creature turned out to
be a sphere. Imagine a sphere passing through a two-dimensional surface (see
the second diagram in the section on hyperspheres for a better idea). This is
another demonstration of how a creature living in two-dimensions would not fully
see a creature in three-dimensions. This pattern continues into the higher
dimensions. [1]
With these patterns in mind, it is interesting to investigate the different
shapes and the patterns among these shapes as we pass from one dimension to
the next. Eulers formula is particularly interesting to investigate as it was
originally only used in the third dimension. In addition, the two most common
four-dimensional shapes are the hypercube and hypersphere. Patterns from the
first, second and third dimensions lead us to conclusions about the fourth
dimension and beyond.
History of the fourth dimension
The fourth dimension is a phenomenon that dates back to as early as the
1800s. Euclids mathematical theories were limited to just the third dimension.
The thought of a fourth dimension intrigued a German man by the name of Georg
Bernard Riemann. Riemann visualized the fourth dimension and believed it
would help with the unification of all physical laws [9]. Riemann made this
phenomenon well known through his lecture entitled On the Hypotheses which
lie at the Bases of Geometry. (Note: the entire lecture was originally in
German.) In this lecture he constantly refers to a multiply extended magnitude;
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in other words, n-dimensions. In his lecture, he also discusses the idea of using
patterns to move from one dimension to the next. He states, Measure consists
in the superposition of the magnitudes to be compared; it therefore requires a
means of using one magnitude as the standard for another.[12] After this
lecture, the fourth dimension soon appeared in art, literature and was a key
aspect in the cubist revolution [9]. Riemanns lecture stirred up society and lead
to much more discussion and investigation regarding the fourth dimension and
beyond.
In the early 1900s Albert Einstein tinkered with the idea of the fourth
dimension but did not put it in writing. When he stumbled across Riemanns
lecture, he found a way to put it to use. Using the fourth dimension Einstein
simplified his equations for gravity and was able to come up with an exact
formulation of the General Principle of Relativity. In his theory of relativity, he
refers to a four-dimensional space-time continuum [4] using time as the fourth
dimension. Albert Einsteins Relativity: The Special and General Theory states,
we must regard x1, x2, x3as space co-ordinates and x4as a time co-ordinate.
[4] Straying from Euclidean geometry increased the preciseness of Einsteins
theory.
Today the fourth dimension is used daily in calculations to explain our
universe, however it is not tangible, no one has seen it or felt it (except in
dreams), and there are varying opinions as to whether it represents time or a
direction perpendicular to all directions in normal space(tangible directions).
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Beyond Eulers formula
It is fair to say that four-dimensional figures and their properties are
derived from their two and three-dimensional counterparts. Around 1750 Euler
discovered a formula for platonic solids or regular three-dimensional figures:
Vertices Edges + Faces = 2. During his time, Euler was unable to correctly
prove that his formula was true. It was not proven until 1794 by Legendre [7].
From that point forward, this formula was used as a basis for other
mathematicians and other formulas. By tweaking Eulers formula, it is easy to
see a pattern from dimension to dimension. In two-dimensions, for instance, the
formula changes to Vertices Edges = 0. For future reference V vertices, E-
edges, F-faces and C-cells. Vertices, edges and faces are common to our three
dimensional figures. Four-dimensional figures and higher contain three-
dimensional units which are called cells. See table below for two-dimensional
examples:
Regular
Polygons
V E V - E
Triangle 3 3 0
Square 4 4 0
Pentagon 5 5 0
This table could continue to include hexagons, heptagons, octagons, etc. The
results would be the same. Notice that in two dimensions there are only two
variables, vertices and edges; because in two dimensions the number of faces
on a regular polygon is one. In three dimensions as stated previously, the
formula is V E + F = 2. See the table below for examples [2].
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Regular 3D Figures V E F V-E+F
Cube 8 12 6 2
Tetrahedron 4 6 4 2
Octahedron 6 12 8 2
Dodecahedron 20 30 12 2
Icosahedron 12 30 20 2
Notice in three dimensions there are now three variables of importance:
vertices, edges and faces. In four dimensions the formula is V E + F C = 0.
See the table below [3]:
Regular 4D
Figures
V E F C V-E+F-C
Hypercube 16 32 24 8 0
16-Hedroid 8 24 32 16 0
24-Hedroid 24 96 96 24 0
Notice in four dimensions, cells are added to the list of variables. There are six
regular 4D figures that follow the formula in the table above. As the dimension
increases by one, the number of variables of importance also increases by one.
Also, notice that in even dimensions the formulas give a result of zero and in odd
dimensions the formulas give a result of 2.
A Swiss man by the name of Ludwig Schlfli was the first to generalize the
pattern observed above and Jules Henri Poincar was the first to prove it. Let
A1, A2, A3, and A4represent vertices, edges, faces and cells respectively. The
table below summarizes the patterns previously observed [15].
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1D A1= 2
2D A1 A2= 0
3D A1 A2+ A3= 2
4D A1 A2+ A3A4= 0
Notice that the odd dimensions end in addition and the even dimensions end in
subtraction. Also, notice that the subscript of the last A corresponds to the
dimension.
Therefore, the formula for N-dimensions is as follows:
ND: A1 A2+ A3+ (-1)n 1 *An or 1- (-1)
n [15]
Corresponds to the n-dimension. Note:as dimension increases by one sodoes the number of variables
Determines addition orsubtraction. Commonly usedin sequences and/or series.
With this formula, it is easy to project the Euler Characteristic for any dimension.
Eulers formula is also commonly used today in graph theory.
Characteristics of the hypercube
Let us begin with a basic two-dimensional shape: a square. A two-
dimensional square has a length and a width. These dimensions are
perpendicular to each other (i.e. x-axis and y-axis). When extended into the third
dimension, the square becomes a cube with a length, width and height. Again,
these dimensions are all perpendicular to each other (i.e. x-axis, y-axis and z-
axis). When a cube is extended into the fourth dimension, it translates along a
path perpendicular to the three existing dimensions. A hypercube (otherwise
known as a tesseract) contains eight cubes. According to Clifford A. Pickover,
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For a tesseract, the eight cubes are: the large cube, the small interior cube and
six hexahedrons (distorted cubes) surrounding the small interior cube. [11] It
may be easier to imagine the net of a hypercube, as compared to the net of a
cube.
HypercubeCubeNote: You can only foldthe net of hypercube inthe fourth dimension.
It is easy to visualize how the net of a cube folds into a cube; however, it is
difficult to visualize the net of a hypercube folding into a hypercube. The reason
for this is that in the fourth dimension objects can pass through surfaces without
breaking the surface. Therefore, you must imagine that the net of the hypercube
is being folded in the fourth dimension. See the color-coded diagrams below to
help visualize how the net of a hypercube would be folded in the fourth
dimension. The colors and diagrams correspond to each other. Again, keep in
mind that you can only fold this into a hypercube in the fourth dimension.
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[13]
Moreover, a cube contains 8 vertices, 12 edges and 6 planes. When
translated into the fourth dimension it contains 8 additional vertices, 12 additional
edges as well as 8 edges created by connecting the vertices of the two cubes
respectively. In addition, it contains 6 additional planes, as well as 12 new
planes traced by the 12 line segments. This gives us a total of 16 vertices, 32
edges and 24 planes [8]. Furthermore, according to Henry Manning:
the new figure will have a cube at the beginning of the movement and
another cube at the end, and in addition each of the six squares bounding
the original cube will by their movement trace a new cube, thus adding six
new cubes to the two already mentioned, or eight cubes in all[8]
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Compare the two-dimensional sketches below.
Hypercube
Cube
Note: Be sure totake a good look!
Cubes aretranscending in alldifferent directions.
These sketches represent a cube and a hypercube being projected onto a two-
dimensional plane. In additions, the shadow of a three dimensional figure is two
dimensional, therefore the shadow of a four dimensional figure is three-
dimensional. The picture below is of a three-dimensional figure that represents
the shadow of a hypercube at a particular moment in time.
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Through understanding of the basic concepts that lead to the hypercube, deriving
the volume formula is very simple.
Given the commonly used perimeter, area and volume formulas for
squares and cubes, it is very easy to see a pattern that extends into the fourth
dimension. The table below gives us information regarding two, three and four
dimensional squares, cubes and hyper-cubes respectively.
Square Cube Hypercube
Perimeter P= 4s
Area A = s2 SA = 6s2
Volume/Hyper-
surface area
V=s3 HSA=8s3
Hyper-volume HV=s4
* s represents the length of a side
Patterns exist down the diagonals of the table. The hyper-surface area of a
hypercube is 8s3. A formula can be derived to find the perimeter, surface area
and hyper-surface area etc That is 2n(s)n-1
where n represents dimension. Itfollows that in the 5th dimension the hyper-hyper surface area of a hyper-hyper
cube would be HHV=10s4. A formula can also be derived to find the area,
volume, hyper-volume etc. That formula is sn, again where n represents
dimension [11]. Keep in mind, however, that the hyper-cube in the fourth
dimension collapses onto itself, and therefore the surface area and volume
formulas following from the previous dimensions, depend on the shape of the
figure at a particular moment in time. These patterns from dimension to
dimension lead us to shapes that are more complicated.
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Characteristics of the Hypersphere
Hyperspheres are more difficult to imagine than hypercubes. It is easiest
to picture a hypersphere by again looking at the patterns. First, imagine the
cross sections of a hollow circle passing through a line [7].
Notice that you would first see a point, then two points gradually getting further
apart, then gradually getting closer together until they coincide back into one
point. Now imagine a three dimensional sphere passing through a plane [7].
Note: The gray line in this diagramrepresents a plane in 3-space.
Notice that the cross sections of a sphere consist of different size circles. The
outer circles are solid and the inner circles are hollow [7].
In conclusion the two dimensional circle is broken into one-dimensional
points and the three dimensional sphere is broken into two-dimensional circles.
Therefore it is apparent that a four dimensional hypersphere, when passing
through a three dimensional surface, will yield three-dimensional spheres of
varying sizes. Some of which will be solid and some of which will be hollow.
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Another pattern to examine is the number of points it takes to create a circle,
sphere and hypersphere. It takes 3 non-collinear points to create a circle, 4 non-
coplanar points to create a sphere, and therefore 5 points not in the same
space to create a hypersphere [6].
Algebraically it is easy to determine the equation for a hypersphere. The
algebraic equation for a circle in two dimensions is (x-h)2+(y-k)2=r2, where (h, k)
represents the center point and rrepresents the radius. The equation for a
sphere in three dimensions is (x-h)2+(y-k)2+(z-l)2=r2, where (h, k, l) represents the
center point and rrepresents the radius. Therefore in four dimensions it is easy
to see that the algebraic equation for a hypersphere is (x-h)2+(y-k)2+(z-l)2+(w-
m)2=r2 where (h, k, l, m) represents the hyper-center and rrepresents the hyper-
radius.
The hypervolume formula for a hypersphere can also be determined by
examining the area of a circle and the volume of a sphere. The area formula for
a circle is often used and is A= r2. The volume formula for a sphere is V= 3
3
4r .
From looking at these two formulas, it is difficult to determine a pattern that
extends into the fourth dimension. It is helpful to look at the derivation of the
area and volume formulas in order to extend a formula into the fourth dimension.
The following derivations assume that the center of all of the circles, spheres and
hyperspheres is at the origin; therefore, we can eliminate the variables h, k, l and
m.
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Area of a Circle Derivation:
222ryx =+ Equation of a circle
22xry =
Solve for y
2=A dxxrr
r
22
Twice the area from r to r
=2
2
2
1
222cos)sin(2
drrrA
Let sinrx =
2
2
cos
==
==
=
rx
rx
drdx
=2
2
2
1
22 cos)sin1(2
drA
=2
2
22cos2
drA
Trigonometric Identity
1cossin22 =+
2
2
22sin
4
1
2
12
+= rA
Table of Integrals [14]
+
+
= )sin(4
1
22
1
sin4
1
22
1
22
rA
Substitution
=
442
2 rA
Simplify
A= 22
22 rr
=
Two Dimensional Circle AreaFormula
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Using this same method we can find the formula for the volume of a sphere [2].
2222rzyx =++
2222zryx =+
Therefore the radius of the 2D equation
22
zr =
Equation of a sphere
)(22 zrA = Using formula
derived for the areaof a circle.
=2
2
22)(
dzzrV
Let:
2
2
cos
sin
==
==
=
=
rz
rz
drdz
rz
=2
2
222cos)sin(
drrrV
Substitution
=2
2
23cos)sin1(
drV
Distributive Property
=2
2
33cos
drV
TrigonometricIdentity
( )2
2
23sincos2
3
1
+= rV
Table of Integrals[14]
+
+=
2sin
2cos2
3
1
2sin
2cos2
3
1 223 rV
( )( ) ( )( )
+
+= 102
3
1102
3
13rV
Simplify
3
3
4rV =
Three DimensionalVolume of a SphereFormula
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Therefore, using the above formulas the hyper-volume of a hyper-sphere can be
derived in the same manner [2].
22222 rzyxw =+++
22222 zryxw =++
Therefore the radius of the 3D equation 22 zr =
Equation of aHypersphere
( )3223
4zrV =
Using the formula forthe volume of a sphere.
( )
=r
r
dzzrHV 23
22
3
4
Let:
2
2
cos
sin
==
==
=
=
rz
rz
drdz
rz
( )
=2
2
23
222cossin
3
4
drrrHV
Substitution
( )
=2
2
23
23cossin1
3
4
drrHV
Distributive Property
( )
=
2
2
23
24 coscos34
drHV
Pythagorean Identity
=2
2
44cos
3
4
drHV
Simplify
+=
2
2
234cos
4
3sincos
4
1
3
4
drHV
Table of Integrals [14]
++=
2
2
342sin
4
1
2
1
4
3sincos
4
1
3
4
rHV
Table of Integrals [14]
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+
+
+
+
=
22sin
4
1
22
1
4
3
2sin
2cos
4
1
22sin
4
1
22
1
4
3
2sin
2cos
4
1
3
4
3
3
4
rHV
4
3
4rHV =
16
3
16
3
Simplify
2
42r
HV
= Four dimensionalhypervolume of ahypersphere
In looking at the patterns moving from the second dimension to the fourth
dimension, a formula can be derived:( )!
2
2
n
rV
nn
= [10] where r represents the
radius and n represents an even dimension. The formula for figures in odd
dimensions gets tricky as we have only derived the formula for the third
dimension and are at this point unable to determine a pattern. Keep in mind that
figures in the fourth dimension are constantly on the move and therefore the
mathematical formulas only hold true for one particular moment in time. With
that in mind, we could potentially find the volume of a figure in 10-D, 100-D,
1000-D, 10000-D etcbut notice that as n increases to such large values the
Volume gets decreasingly smaller. The reason for this is:n
lim( ) !
2
2
n
rn
n
= 0. This
shows that as the dimensions gradually increase, the volume first increases, then
reaches a maximum and gradually decreases to a horizontal asymptote of zero.
A hypersphere consists of an infinite number of spheres able to move
around each other, over each other and pass through each other. Hyperspheres
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are more difficult to imagine than hypercubes, however they are very interesting
figures.
Summary
In conclusion, the patterns that exist from one dimension to the next make the
study of higher dimensions very intriguing. Euler just saw a pattern in common
three-dimensional shapes that eventually turned into a multi-dimensional
phenomenon. Hypercubes and hyperspheres, although difficult to visualize,
follow the same logic as squares and circles, and as cubes and spheres. These
multi-dimensional figures and ideas have been seen in cubist paintings, used in
literature, and have assisted in proving mathematical theories. The fourth
dimension and beyond has greatly affected our society.
Acknowledgment
The funding for this project was provided by the National Science Foundation.
Special thanks to my advisor, Dr. Aimee Ellington, for all of her support through
the entire process. In addition, thank you to my committee members, Dr.
Rueben Farley and Dr. William Haver.
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References
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