Pascal's Triangle

Pascal’s Triangle is a famous triangular array of numbers with several interesting properties.  Each line of the triangle starts and ends with the number 1, and the numbers in between can be obtained by finding the sum of the two numbers above it.  The triangle can then be continued indefinitely.

                            

						1
					1		1
				1		2		1
			1		3		3		1
		1		4		6		4		1
	1		5		10		10		5		1
1		6		15		20		15		6		1

Pascal’s Triangle

Unfortunately, Pascal’s triangle is rarely mentioned in high school math curriculum, except as a footnote for expanding binomials, because the coefficients for the expanded form of (x + y)ⁿ match up with the rows of Pascal’s Triangle.  For example, (x + y)² = x² + 2xy + y² (1, 2, 1), (x + y)³ = x³ + 3x²y + 3xy² + y³ (1, 3, 3, 1), and (x + y)⁴ = x⁴ + 4x³y + 6x²y² + 4xy³ + y⁴ (1, 4, 6, 4, 1).  But there are many other unique properties to the triangle.

First of all, the rows of Pascal’s triangles add up to a power of 2.  For example, the numbers in the third row (1, 2, 1) add up to 4 which is 2², the numbers in the fourth row (1, 3, 3, 1) add up to 8 which is 2³, and the numbers in the fifth row (1, 4, 6, 4, 1) add up to 16 which is 2⁴.

						1
					1		1
				1		2		1
			1		3		3		1
		1		4		6		4		1
	1		5		10		10		5		1
1		6		15		20		15		6		1

The rows of Pascal’s Triangle add up to powers of 2.

Second, the diagonals of Pascal’s Triangle represent totals for different simplexes (lines, triangles, tetrahedrons, etc.).  For example, the third diagonal row (1, 3, 6, 10, 15, etc.) are triangular numbers, because objects in these amounts can form equilateral triangles, and the fourth diagonal row (1, 4, 10, 20, etc.) are tetrahedral numbers, because of objects in these amounts can form tetrahedrons (triangular pyramids).

						1
					1		1
				1		2		1
			1		3		3		1
		1		4		6		4		1
	1		5		10		10		5		1
1		6		15		20		15		6		1

The diagonals of Pascal’s Triangle represent totals for different simplexes.

Triangular Numbers: 1, 3, 6, 10, etc.

Tetrahedral Numbers: 1, 4, 10, 20, etc.

Third, another diagonal of a different slope in the numbers of Pascal’s triangle add up to different numbers in the Fibonacci sequence.  As you might recall, the numbers in the Fibonacci sequence can be obtained by adding the two numbers before it, so starting with two ones, it is 1, 1, 2, 3, 5, 8, 13, etc.  Numbers in the pictured diagonals (and parallel diagonals) of Pascal’s triangle add up to the numbers in the Fibonacci sequence.  For example, 1 = 1, 1 = 1, 1 + 1 = 2, 1 + 2 = 3, 1 + 3 + 1 = 5, 1 + 4 + 3 = 8, 1 + 5 + 6 + 1 = 13, etc.

						1
					1		1
				1		2		1
			1		3		3		1
		1		4		6		4		1
	1		5		10		10		5		1
1		6		15		20		15		6		1

These Pascal Triangle diagonals add up to different numbers in the Fibonacci sequence.

Fourth, the numbers in the Pascal’s triangle represent the number of combinations that are possible in a set, which is useful in statistics.  For example, if you wanted to know how many different possible two-card hands you can be dealt from a deck of five cards, it would be ₅C₂ = 10, which is 5 + 1 = 6^th row, and 2 + 1 = 3^rd number in Pascal’s Triangle.  (If the five cards are A, B, C, D, and E, then all the possible two-card hands are AB, AC, AD, AE, BC, BD, BE, CD, CE, and DE, which makes ten total possibilities.)  The number of different possible four-card hands you can be dealt from a deck of six cards would be ₆C₄ = 15, which is the 6 + 1 = 7^th row, and 4 + 1 = 5^th number in Pascal’s Triangle.

0C0

1C0

1C1

2C0

2C1

2C2

3C0

3C1

3C2

3C3

4C0

4C1

4C2

4C3

4C4

5C0

5C1

5C2

5C3

5C4

5C5

6C0

6C1

6C2

6C3

6C4

6C5

6C6

Combination Sets in Pascal’s Triangle

Finally, as mentioned in an earlier blog post, the number of endpoints, line segments, faces, and solids of each multi-dimensional simplex corresponds to a row in Pascal’s Triangle.  For example, 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); the 3-D simplex (tetrahedron) is comprised of 4 endpoints, 6 line segments, 4 faces, and 1 solid (4, 6, 4, 1), etc.

3D Simplex: Tetrahedron

Comprised of 4 endpoints, 6 line segments, 4 faces, and 1 solid

Pascal’s Triangle has some amazing unique properties.  It can be applied to the expansion of binomials, powers of two, sequences of numbers, statistics, and properties of simplexes.  It truly is a wonder in mathematics!

Multi-Dimensional Shapes and Pascal's Triangle

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.

                            

						1
					1		1
				1		2		1
			1		3		3		1
		1		4		6		4		1
	1		5		10		10		5		1
1		6		15		20		15		6		1

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.