The simplest
geometric object is a point. In
Geometry, we learn that a point is a single location without length, width, or
height, and is usually represented by a dot.
Euclid defined the point as “that which has no part”. Because it has neither length, width, nor
height, it is a zero-dimensional object.
Point
The next
simplest geometry object is the line. A
line is straight, infinitely long, and has no thickness. Euclid said that a line has “breadthless
length”. A line segment is part of a
line that begins at a point and ends at another point. Both lines and line segments have length, but
neither width nor height, and so are one-dimensional objects.
Line Segment
If a point
is zero-dimensional because it has neither length, width, nor height, and a
line or line segment is one-dimensional because it has length but neither width
nor height, then it follows that an object with length and width but no height
is two-dimensional (such as a triangle, square, rectangle, circle, etc.), and
an object with length, width, and height is three-dimensional (such as a
pyramid, cube, sphere, etc.)
Let’s
identify the simplexes of each dimension, or the object in each dimension that
is made entirely of straight line segments and the least amount of
endpoints. The simplex of zero
dimensions is a point (since that is the only zero-dimensional object). The simplex of one dimension is a line
segment, which has two endpoints. The simplex
of two dimensions is a triangle, because it has the least amount of endpoints
(three) of all two-dimensional shapes with straight sides. Finally, the simplex of three dimensions is
the tetrahedron (a triangular-based pyramid) because it has the least amount of
endpoints (four) of all three-dimensional objects with straight sides.
2D Simplex: Triangle
3D Simplex: Tetrahedron
If we
examine the number of endpoints, line segments, faces, and solids of each
simplex, we will observe some familiar numbers.
The 0-D simplex (a point) is comprised of 1 endpoint (1). The 1-D simplex (line segment) is comprised
of 2 endpoints and 1 line segment (2, 1).
The 2-D simplex (triangle) is comprised of 3 endpoints, 3 line segments,
and 1 face (3, 3, 1). And the 3-D
simplex (tetrahedron) is comprised of 4 endpoints, 6 line segments, 4 faces,
and 1 solid (4, 6, 4, 1). We can see
that the number of endpoints, line
segments, faces, and solids of each multi-dimensional simplex corresponds to a
row in Pascal’s Triangle, a famous triangular array of numbers in which
each number is the sum of the two numbers above itself.
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Pascal’s Triangle
Using
Pascal’s Triangle we can extrapolate properties of higher dimensional
simplexes. A 4-D simplex (sometimes
called a pentachoron or a hyper-pyramid) would have 5 endpoints, 10 line
segments, 10 faces, 5 solids, and 1 four-dimensional object (5, 10, 10, 5, 1),
and a 5-D simplex would have 6 endpoints, 15 line segments, 20 faces, 15
solids, 6 four-dimensional objects, and 1 five-dimensional object (6, 15, 20,
15, 6, 1), and so on.
4D Simplex: Pentachoron
It is
difficult to picture an object in four-dimensions since we live in a
three-dimensional space, but we can try to wrap our minds around it if we
observe the progression of simplexes in each dimension. One such progression is that placing a new
endpoint in the exact center of all the other existing endpoints and attaching
all the endpoints with line segments gives an aerial view of the next simplex
in the next dimension. For example,
placing the midpoint on a line segment (1-D simplex) gives an aerial view of an
upright triangle (2-D simplex), or placing a point in the centroid of a
triangle (2-D simplex) and attaching all the endpoints with line segments gives
an aerial view of a tetrahedron (3-D simplex).
Aerial View of a Tetrahedron
In the same
way, placing a point in the centroid of a tetrahedron (3-D simplex) and
attaching all the endpoints with line segments should give some sort of aerial
view of a pentachoron (4-D simplex).
This progression also explains why properties of simplexes correspond
with the numbers in Pascal’s Triangle, because the new point in the center of
the simplex in k dimensions creates the same amount of n-dimensional objects as
existing (n – 1)-dimensional objects for the simplex in (k + 1)
dimensions. For example, the new point
at the centroid of a triangle creates three new faces that correspond to the
three existing edges, along with the existing one face, which means that the
tetrahedron will have 3 + 1 = 4 faces.
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