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POLYGONAL NUMBERS AND THE
PYTHABACUS |
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The Pythabacus uses the properties of the polygonal numbers: triangles, squares and oblongs
to reveal geometric solutions to arithmetic and algebraic problems. The Pythagoreans
explored polygonal numbers by arranging stones, while the Pythabacus
achieves such exploration with a patented bead and rod configuration. Polygonal numbers are the sums of the numbers of stones necessary to build a sequence of polygons with (s) sides, with all the polygons in the sequence having one common vertex. To display polygons click a gold bead on the Pythabacus. Press Enter to Delete Polygons! To change number of sides place cursor in text box and delete current number. Then enter new number. The number of additional stones added to build each successive polygon compose an arithmetic sequence. Each nth term in such a sequence will equal 1 + (n-1)(s-2), where (s) equals the number of sides composing the polygons of the sequence. The middle box to the right of the Pythabacus displays the nth term. The sum of a sequence P(s,n) for s=3 at the nth term will equal n(n+1)/2. In general, P(s,n) for an s-polygon will equal [P(3,n-1)] times [s-2] plus [n] or ((n(n-1)/2)(s-2))+n. On the Pythabacus, P(3,n-1) will always equal the triangle of brown beads above a (n) number of gold beads. The bottom box to the right of the Pythabacus displays P(s,n), the sum of the sequence. P(4,n) or a square equals [2] times [P(3,n-1) plus [n] or n(n-1)+n. This logically and algebraically means that every square is composed of two P(3,n-1) triangles plus a line of n units. This is demonstrated on the Pythabacus when you separate an even number (n) of gold beads into 2 equal groups: (a) and (b), where (a) equals (b), you will produce between (a) and (b) a quadrilateral with a number of beads equal a square number of beads or equal (a times b). It is this property of the Pythabacus that allows the exploration of geometric solutions to arithmetic and algebraic problems. Click Link to Explore More about Polygonal Numbers |