Angela Wood

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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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Let's Examine the Patterns!
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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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A Fly in Zero Dimensions!

• 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.

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A Fly on One Dimension

• 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

• 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

• 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?

• 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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Whats the Point?

• 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.

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FlatlandBy 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!!

• 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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Eulers Formula

• States a mathematical relationship between the
  number of vertices, edges and faces of a
  polyhedron.
• In 3-dimensional space,
• vertices-edgesfaces2
• Or V-EF2

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Eulers Formula in 4-D
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N-Dimensional Polyhedron

• Schlaflis Formula
• N0-N1N2-(-1)n-1Nn-11-(-1)n-1
• I still need to do additional research to fully
  understand this formula, however
• for even dimensions (2,4,6),
• N0-N1N2-N30
• and for odd dimensions (3,5,),
• N0-N1N2 -2
• as shown in the previous charts.

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This is Basically the Research I Have Completed
Up To This Point.
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Future Research

• I plan on fully researching Eulers formula and
  Schlaflis 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

• 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

• 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 Press1999
• Polyhedral Formula
• http//mathworld.wolfram.com/PolyhedralFormula.ht
  ml
• Regular Polyhedra
• http//www.cut-the-knot.com/do_you_know/polyhedra
  .shtml