characteristics in the fourth dimension

8/2/2019 Characteristics in the Fourth Dimension

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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


++=

2

2

342sin

4

1

2

1

4

3sincos

4

1

3

4

rHV


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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

[1] Abbott, Edwin A. (1884) Flatland: A romance of many dimensions.

(Aloysius West, transcribed) Erik Max Francis. Internet:

http://www.alcyone.com/max/lit/flatland.

[2] Baragar, Arthur. (2000). A Survey of Classical and Modern Geometries

with Computer Activities. Prentice Hall

[3] Coxeter, H.S.M. (1969) Introduction to Geometry. (2nd ed.) New York:

John Wiley and Sons

[4] Einstein, Albert (1920). The Space-Time Continuum of the General Theory

of Relativity is not a Euclidean Continuum. In Relativity: The Special and

General Theory. New York: Henry Holt, Bartleby.com2000. Internet:

www.bartleby.com/1731

[5] Eppestein, David, Seventeen Proofs of Eulers Formula: V-E+F=2,

Internet:http://www.ics.uci.edu/~eppstein/junkyard/euler.

[6] Fitch, Graham Denby. (1960) An Elucidation of the Fourth Dimension. In

Henry P. Manning, The Fourth Dimension: Simply Explained. (pp.43 -51).

New York: Dover Publications, Inc.

[7] Fuquay, Jeff. The Hypersphere, Internet:

http://www.geocities.com/jsfhome/Think4d/Hyprsphr/hsphere.html .

[8] Gunnel, Leonard C. (1960). Length, Breadth, Thickness, and Then What?.

In Henry P. Manning, The Fourth Dimension: Simply Explained. (pp. 110

117). New York: Dover Publications Inc.

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http://www.bartleby.com/1731http://www.geocities.com/jsfhome/Think4d/Hyprsphr/hsphere.htmlhttp://www.geocities.com/jsfhome/Think4d/Hyprsphr/hsphere.htmlhttp://www.bartleby.com/1731


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[9] Lesperance, Ian. The Fourth Dimension. Internet:

http://www.elliterate.com/archives/elliterate03/fourthdimension.htm

[10] Manning, H. P. (1928). Geometry of Four Dimensions. New York: The

MacMillan Company.

[11] Pickover, Clifford A. Surfing Through Hyperspace: Understanding Higher

Universes in Six Easy Lessons. New York: Oxford University Press.

[12] Riemann, Bernhard. (1867). On the Hypotheses Which Lie at the Bases of

Geometry. (W.K. Clifford, Tran.). Internet:

http://www.maths.tcd.ie/pub/HistMath/People/Riemann/Geom/WKCGeom.

html

[13] Rucker, Rudy. (1984) The Fourth Dimension: A Guided Tour of the Higher

Universes. Boston: Houghton Mifflin Company

[14] Stewart, James (2001). Calculus: Concepts and Contexts. (2nd ed.).

California: Brooks/ Cole.

[15] Weisstein, Eric W. Polyhedral Formula. From Math WorldA Wolfram

Web Resource. Internet:

http://mathworld.wolfram.com/PolyhedraFormula.html .
http://www.elliterate.com/archives/elliterate03/fourthdimension.htmhttp://www.maths.tcd.ie/pub/HistMath/People/Riemann/Geom/WKCGeom.htmlhttp://www.maths.tcd.ie/pub/HistMath/People/Riemann/Geom/WKCGeom.htmlhttp://mathworld.wolfram.com/PolyhedraFormula.htmlhttp://mathworld.wolfram.com/PolyhedraFormula.htmlhttp://www.maths.tcd.ie/pub/HistMath/People/Riemann/Geom/WKCGeom.htmlhttp://www.maths.tcd.ie/pub/HistMath/People/Riemann/Geom/WKCGeom.htmlhttp://www.elliterate.com/archives/elliterate03/fourthdimension.htm

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