Visualizing the Fourth Dimension

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Visualizing the Fourth Dimension
Steven R. Lay Department of Natural Science and
Mathematics Lee University slay_at_leeuniversity.edu
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Building a 4-Dimensional Simplex
A 0-dimensional simplex is a single point.
Add a point x2 that is not coincident with x1.
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A 1-dimensional simplex is a line segment.
x2
Join x2 to x1 by a straight line segment.
x1
Note Each endpoint of a 1-dimensional simplex
is a 0-dimensional simplex.
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A 2-dimensional simplex is a triangle.
Add a point x3 that is not collinear with x1 and
x2.
Join x3 to each of the points in the line segment.
x2
x1
Note Each edge of a 2-dimensional simplex is a
1-dimensional simplex.
There are 3 edges1 from the original line
segment and 2 from
joining x3 to each of the endpoints x1 and x2.
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Add a point x4 that is not in the same plane as
x1, x2, and x3.
x2
x4
x3
x1
Join x4 to each of the points in the triangle.
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A 3-dimensional simplex is a tetrahedron.
x2
x4
x3
x1
Note Each face of a 3-dimensional simplex is a
2-dimensional simplex.
There are 4 faces1 from the original triangle
and 3 from joining
x4 to each of the edges of the triangle.
There are 6 edges 3 from the original triangle
and 3 from
joining x4 to each of the vertices of the
triangle.
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Building a 4-Dimensional Simplex
S0
S1
S2
S3
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For a 4-dimensional simplex, S4, we need to add a
point x5 that is not in the 3-space of the
tetrahedron S3 and join x5 to each of the points
in S3.
It will have 5 tetrahedral facets 1 from the
original tetrahedron S3 and 4 from joining
x5 to the four triangular faces of S3.
It will have 10 triangular faces 4 from the
original tetrahedron S3 and 6 from joining
x5 to the six edges of S3.
It will have 10 edges 6 from the original
tetrahedron S3 and 4 from joining x5 to the
four vertices of S3.
x2
x4
x3
x1
S3
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Building a 4-Dimensional Simplex
x2
x4
x3
x1
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The 4-Dimensional Simplex S4
x5
x2
x4
x3
x1
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The 4-Dimensional Simplex S4
x5
x2
x4
x3
x1
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The 4-Dimensional Simplex S4
x5
x2
x4
x3
x1
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The 4-Dimensional Simplex S4
x5
x2
x4
x3
x1
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The 4-Dimensional Simplex S4
x5
x2
x4
x3
x1
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The 4-Dimensional Simplex S4
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Add the point x5 inside the tetrahedron S3.
Then join x5 to all the points in S3.
x4
x5
x2
x1
x3
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x4
x5
x2
x1
x3
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x4
x5
x2
x1
x3
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x4
x2
x1
x3
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x4
x5
x2
x1
x3
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x4
x5
x2
x1
x3
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Building a 4-dimensional Crosspolytope
Select a nonzero point x1 and its opposite ? x1.
Join x1 and ? x1 to form a line segment.
? x1
0
x1
A 1-dimensional crosspolytope X1 is a line
segment symmetric about the origin.
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Select a nonzero point x2 that is not collinear
with x1 and 0.
Join x2 and ? x2 to form a line segment.
Then join all the points in the two line segments
to form a parallelogram.
? x2
? x1
0
x1
x2
A 2-dimensional crosspolytope X2 is a
parallelogram.
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Select a nonzero point x3 that is not coplanar
with x1, x2 and 0.
Join x3 to all the points in X2 to form a pyramid
on the top.
? x2
x3
? x1
x1
x2
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Join ? x3 to all the points in X2 to form
another pyramid on the bottom.
? x2
x3
? x1
x1
x2
A 3-dimensional crosspolytope X3 is an
octahedron.
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Note each edge of X2 generates a triangular face
of X3 when joined to x3 or ? x3. X2 had
4 edges, so there are 2 4 8 triangular faces.
? x2
x3
? x1
x1
? x3
x2
A 3-dimensional crosspolytope X3 is an
octahedron.
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To obtain the 4-dimensional crosspolytope X4,
select a point x4 not in the 3-space of x1, x2,
x3, and 0, and join X3 to x4 and ? x4.
Each triangular face of X3 will generate a
tetrahedral facet of X4 when joined to x4 or ?
x4. Since X3 has 8 triangular faces, X4 will
have 16 tetrahedral facets.
X3
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Select the points x4 and ? x4.
Join these points to X3.
The 4-dimensional crosspolytope X4.
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conv x1, x2, x3, x4
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conv x1, x2, x3, ? x4
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? x2
x3
? x1
x4
x1
? x4
? x3
x2
conv x1, x2, ? x3, ? x4
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conv x1, x2, ? x3, x4
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The 4-dimensional crosspolytope X4.
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Building a Hypercube
(tesseract)
Translate C1 in a direction perpendicular to the
line containing C1.
Then join the translated set to the original set.
A 1-dimensional cube C1 is a line segment.
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Note each edge of a 2-dimensional cube is a
1-dimensional cube.
There are 4 edges 1 from the original line
segment 1 from
the translated line segment 2 from joining
the original vertices to the translated ones.
There are 4 vertices 2 from the original line
segment 2 from the translated line segment
A 2-dimensional cube C2 is a square.
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Translate the square in a direction perpendicular
to the plane containing the square.
Then join the translated set to the original set.
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Note each face of a 3-dimensional cube is a
2-dimensional cube (square).
There are 6 faces 1 from the original square,
1 from the
translated square, 4 from joining the
original edges to the translated ones.
There are 12 edges 4 from the original
square, 4 from the translated square, 4
from joining the original and
translated vertices.
There are 8 vertices 4 from the original
square, 4 from the translated square,
A 3-dimensional cube C3 is a cube.
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To get a 4-dimensional hypercube C4, translate
the cube in a direction not in the 3-space of the
cube and join the original and translated cubes.
It will have 8 cubic facets 2 from the
original and translated cubes, 6 from joining
the original and translated faces of C3.
It will have 24 square faces 6 from the
original cube, 6 from the translated cube,
12 from joining the original and translated edges
of C3.
It will have 32 edges 12 from the original
cube, 12 from the translated cube, 8 from
joining the original and translated vertices of
C3.
It will have 16 vertices 8 from the
original cube, 8 from the translated cube.
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The 4-Dimensional Hypercube
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The 4-Dimensional Hypercube
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The 4-Dimensional Hypercube
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The 4-Dimensional Hypercube
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The 4-Dimensional Hypercube
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The 4-Dimensional Hypercube
Place the translated cube inside the original
cube.
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Unfolding a Square
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Unfolding a Cube
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Unfolding a Hypercube
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Is the Real World 4-Dimensional?
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John 2019 That evening, on the first day of the
week, the disciples were meeting behind locked
doors because they were afraid of the Jewish
leaders. Suddenly, Jesus was standing there
among them! "Peace be with you," he said.
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Crucifixion (Corpus Hypercubus) Salvador Dali,
1954