Summation Formulas as Geometrical Dissections

Long ago, the Pythagorian community believed that all things are made out of NUMBER - and for them numbers were real, shapely figures, conceived as arrangements of dots. We call their number-pictures figurate numbers. In the African informal mathematical heritage of ornaments, braids and baskets it's the same: the mathematics is there, in beautiful, visible, touchable ways. I think we make a big mistake when we force our poor modern students of mathematics to run too quickly from such finite, discrete and concrete geometrical number theory to our infinite, continuous, and abstract symbolic modern mathematics. We lose so much that is part of natural intuition and cultural heritage!

In this article, inspired by some work of Henry Chinheya on series, we will try to show how our natural gifts for seeing and touching, weaving, building, carving, fitting together and taking apart, can be used to get a better grasp of how numbers behave. To quote the founder of analytic geometry, René Descartes,

``If we need to apply algebra to geometry and physics, and so bring order and generality to those sciences, we need equally to apply geometry to algebra. Thus, taking the best traits of geometrical analysis and algebra, we may correct the defects of the one by the other. I fear that, without the clarity of geometric figures, algebra becomes a confused obscure art which oppresses the mind, instead of a science which cultivates it.''
First, here is a modern table of some ancient Pythagorian `figurate numbers'. (There are also pentagonal, hexagonal numbers, etc., but we will not use them.) The last two rows are for you to fill in. You will benefit greatly from actually building your numbers out of pebbles or beads or beans, for the first four columns. The numbers in the other columns are best made in 3-dimensions out of wire (or matches) joining berries, say.

In this table, Tn, Sn, Cn, Rn and Ln are triangular, square, cubic, rectangular and L-shaped numbers respectively. Dn, [¯]n and En are the sums, respectively, of the first n triangular, square and cubic numbers. Dn and [¯]n are sometimes referred to as tetrahedral and pyramidal numbers, for reasons which will become obvious as you read on...

Here are some patterns for you to explore, using the table: (answers)

  1. Find a formula for Ln.

  2. Find a nice geometrical dissection showing L2n = 4Ln.

  3. Find a connection between Rn and Cn + Sn, and draw a (3-dimensional) dissection picture to illustrate it.

  4. Find a connection between [¯]n and Sn+Cn+Tn.

Each of the formulas below can be proved, `by mathematical induction' (see the article in Zimaths issue 2.2 page 33) - but it can be seen almost AT A GLANCE in its geometrical representation.

  1. The sum of the first n integers is the nth triangular number; which is half the nth rectangular number. (See diag. 1)
    1+2+3+¼+n = 1
    2
    		 n(n+1) = 1
    2
    		 Rn = Tn

  2. The sum of the first n odd integers is the nth square number. (See diag.2)
    1+3+5+¼+(2n-1) = n2 = Sn

  3. Every odd square is of the form 8T + 1. (See diag. 3)
    S2n+1 = (2n+1)2 = 8 æ
    ç
    è     		n(n+1)
    2
    		 ö
    ÷
    ø     		+1 = 8Tn+1
    (This reminds me of a much deeper result, proved by Gauss and recorded in his diary in coded form: `NUM = \triangle +\triangle +\triangle ' - every number can be expressed as the sum of at most three triangular numbers!)

  4. A square number is twice a triangular number plus the number itself; and it is the sum of two successive triangular numbers. (See diag.4)
    Sn = n2 = 2[1+2+¼+(n-1)]+n = 2Tn-1 + n = Tn + Tn-1
    (Algebraic proof - Use result (1))
  5. A tetrahedral number is a sum of triangular numbers and is a symmetrical sum of isoperimetric rectangles.
    Dn = n
    å
     r = 1 
    		 Tr = 1.n+2.(n-1)+3.(n-2)¼+(n-1).2+n.1

  6. A pyramidal number is the sum of square numbers, and is twice a tetrahedral number plus a triangular number; and is the sum of two successive tetrahedral numbers; and is the sum of a triangular number plus twice sum of isoperimetric rectangles.
    [¯] n = n
    å
     r = 1 
    		 r2 = 2 Dn-1 + Tn = Dn + Dn-1
    In full:
    12+22+¼+n2 = 2[1.(n-1)+2.(n-2)+¼+(n-2).2+(n-1).1]+[1/2]n(n+1)

    This last formula can be proved (of course) by induction. It can also be proved elegantly using results (4) & (5) together, as pointed out by Henry Chinheya.

  7. A pyramidal number is also a sum of trapezia from Tn to n:
    [¯]n
    =
    n. æ
    ç
    è     		n+1
    2
    		 ö
    ÷
    ø     		+(n-1). æ
    ç
    è     		n+2
    2
    		 ö
    ÷
    ø     		+(n-2). æ
    ç
    è     		n+3
    2
    		 ö
    ÷
    ø     		+¼+2. æ
    ç
    è     		n+(n-1)
    2
    		 ö
    ÷
    ø     		+1.n

    Finally, here are two unsolved dissection problems - at least unsolved by me! Perhaps you can see how to `see' them!

  8. 12+22+¼+n2 = [1/6]n(n+1)(2n+1). Can you find a corresponding dissection? Is it possible to fit six pyramids together to make a box of dimensions n×(n+1)×(2n+1)?

  9. Verify from the table that En = Tn2 i.e. 13+23+¼+n3 = [[1/2]n(n+1)]2.
    Find a dissection! (Perhaps it exists in 4-dimensional space!)
- Gavin Hitchcock, with acknowledgements to Henry Chinheya for inspiration.



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On 11 Apr 2001, 15:54.