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The Fourth Dimension. The Fourth Dimension. The Fourth Dimension. The Fourth Dimension. Angela Wood NSF Scholar August 6, 2003. Committee Members. Chairperson: Dr. Aimee Ellington Members: Dr. Rueben Farley Dr. William Haver. Who and When?.

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Angela wood nsf scholar august 6 2003 l.jpg

The Fourth Dimension

The Fourth Dimension

The Fourth Dimension

The Fourth Dimension

Angela Wood

NSF Scholar

August 6, 2003


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Committee Members

Chairperson:

Dr. Aimee Ellington

Members:

Dr. Rueben Farley

Dr. William Haver


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Who and When?

  • Geometry of n dimensions is an early concept that dates back to the 1800s.

  • Scientists, Psychologists, Philosophers as well as Spiritualists also take interest in the 4th Dimension.

  • Mathematicians and Physicists use the 4th dimension in daily calculations to explain our universe.


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What is the 4th Dimension?

  • Some believe it is time.

  • Mathematicians and scientists focus on the fact that it is a direction different from all direction in normal space.



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The First Dimension

An object in the first dimension consists of only one of the fundamental units. For example a line only has length. A line is one dimensional.


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The Second Dimension

The Second Dimension

An object in the second dimension consists of two of the fundamental units. For example, a square has a length and a width. Notice that is you stack lines on top of each other you create a square, or an object in two dimensions.


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The Third Dimension

The Third Dimension

The Third Dimension

An object in the third dimension consists of three of the fundamental units. For example, a cube has a length, width and height. Notice that is you stack squares on top of each other you create a cube, or an object in three dimensions.


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The Fourth Dimension

The Fourth Dimension

The Fourth Dimension

The Fourth Dimension

An object in the fourth dimension consists of four units. For example, a hypercube has a length, width, height and a fourth dimension that is perpendicular to all three of the other units. Look at the a corner of the room and imagine extending a line perpendicular to all three lines at the intersection. If you can visualize stacking cubes into this fourth dimension you create a hypercube.


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How Does this Relate to Us?

Let's Look at a Fly!


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In 0-dimension a fly would be trapped and could not move from one particular point. Imagine a fly trapped in a very small box so that it cannot move in any direction. It has no freedom, or 0 degrees of freedom.

A Fly in Zero Dimensions!


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A Fly on One Dimension from one particular point. Imagine a fly trapped in a very small box so that it cannot move in any direction. It has no freedom, or 0 degrees of freedom.

A fly in one-dimension would only be-able to travel along a line. Backwards and forwards. Imagine a fly trapped in a small tube. The fly could travel forward or backward. It has 1 degree of freedom.


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A Fly in Two Dimensions from one particular point. Imagine a fly trapped in a very small box so that it cannot move in any direction. It has no freedom, or 0 degrees of freedom.

A fly in two dimensions would be able to travel forwards, backwards, left and right. Imagine a fly traveling along a flat surface. The fly now has two degrees of freedom and is traveling in a two dimensional world.


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A Fly in Three Dimensions from one particular point. Imagine a fly trapped in a very small box so that it cannot move in any direction. It has no freedom, or 0 degrees of freedom.

Obviously a fly in three dimensions is able to travel just as it were in our world. That is forwards, backwards, left, right, up and down. What we think of as our world is in three dimensional space.


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What does this have to do with the Fourth Dimension? from one particular point. Imagine a fly trapped in a very small box so that it cannot move in any direction. It has no freedom, or 0 degrees of freedom.

Imagine these flies have blinders on so that they can only see forward.

  • Would the fly in zero-dimension be able to determine what was behind it?

  • Would the fly in one-dimension be able to determine what was to the left and right of it?

  • Would the fly in two-dimensions be able to see what was above and below it?


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What’s the Point? from one particular point. Imagine a fly trapped in a very small box so that it cannot move in any direction. It has no freedom, or 0 degrees of freedom.

Therefore, would we, living in a three dimensional world be able to see beyond our three dimensions?

From this, it is feasible that a fourth dimension exists that we cannot see.

This will take a lot of imagination so please bare with me.


Flatland by edwin a abbott l.jpg
Flatland from one particular point. Imagine a fly trapped in a very small box so that it cannot move in any direction. It has no freedom, or 0 degrees of freedom.By Edwin A. Abbott

View from above.

  • Flatland is a book written in 1884 that describes the phenomenon we just looked at.

  • Flatland is a world of two dimensional creatures. The towns consists of triangles, squares, pentagons etc… The more sides a “person” has the more important they are in society. A circle is the most important figure in their society.

  • In flatland, all the creatures can see are lines and points. Nothing has a height.

  • Imagine being a caterpillar who can only see straight forward. This is how this entire society lived until one day….

View from flatland.


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A Creature From Space-land Visits!! from one particular point. Imagine a fly trapped in a very small box so that it cannot move in any direction. It has no freedom, or 0 degrees of freedom.

  • In Chapters 15 and 16 of Flatland a creature form space-land comes to visit.

  • The flatland creatures do not understand how this creature gets smaller and larger.

  • This space-land creature turns out to be a sphere.

  • Imagine a sphere passing through a plane. The circle that was visible to the flatland creature increased and decreased in size.

  • Again, this explains how creatures in two-dimensional space would not be-able to fully see a creature from three-dimensional space.


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States a mathematical relationship between the number of vertices, edges and faces of a polyhedron.

In 3-dimensional space,

vertices-edges+faces=2

Or V-E+F=2

Euler’s Formula


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Euler’s Formula in 4-D vertices, edges and faces of a polyhedron.


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N-Dimensional Polyhedron vertices, edges and faces of a polyhedron.

Schlafli’s Formula

N0-N1+N2-…+(-1)n-1Nn-1=1-(-1)n-1

I still need to do additional research to fully understand this formula, however

for even dimensions (2,4,6…),

N0-N1+N2-N3+…=0

and for odd dimensions (3,5,…),

N0-N1+N2 -…=2

as shown in the previous charts.



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Future Research Point.

  • I plan on fully researching Euler’s formula and Schlafli’s formula in order to relate them to the fourth dimension.

  • This will extend to looking in depth at polyhedroids, their nets and angles.

  • I would like to focus my project on these figures and work some proofs of the corresponding formulas.

  • Ideally I will be able to prove or disprove the idea of the Fourth Dimension, however this has been underway for several years and not yet been accomplished.


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Estimated Dates Point.

Research Completed: April 2004

First Draft for Review: May 2004

Final Draft: June/July 2004

Submit for Publication: July 2004

Present at Greater Richmond Math Council:

Fall 2004


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References Point.

Baragar, Arthur A Survey of Classical and Modern Geometries with Computer Activites.

Coxeter, H.S.M. Introduction to Geometry: 2nd Edition. John Wiley and Sons: 1969

Pickover, Clifford A. Surfing Through Hyperspace. Oxford University Press:1999

Polyhedral Formula

http://mathworld.wolfram.com/PolyhedralFormula.html

Regular Polyhedra

http://www.cut-the-knot.com/do_you_know/polyhedra.shtml


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