====== imprecise precision: using grids for polygons ======
Grids can be used to embody exactness of mathematical artifacts without having the precision of pencil and ruler skills--concepts of precise lines without reality of line widths or sloppiness getting in the way. 

===== Sacrificing Regularity ===== 
Here are some examples of polygons where only the square is regular. 
^ concept ^ description ^ details ^
^ precision | Using grid points to create polygons. These examples contain angled lines with small integer (1 or 2) x & y components. A focus could be on developing descriptions of non-horizontal and non-vertical lines as. "so many up (down) so many right (left)."  | {{:poly1.svg?400x400|}} |
^ midway points | To make better use of grids, midway points can be used to either simply make smaller polygons or debatably improve upon the regularness of polygons | {{:poly2a.svg?400x400|}}  |
^ simple spirals | Use grid polygons to make spirals. The angled lines help to ensure lines of the spirals' lines remain parallel. Each of the straight lines lengths is a multiple of the original polygon's side. There is a pattern to the multipliers of the originals. For the triangle, the multipliers follow this sequence: 1, **2**, 3, 4, **5**, 6,... The multipliers 2 and 5 (bold in the sequence) are two and five times the original horizontal side not to be confused with the lengths of the sides 4 and 10. Similarly, the diagonal sides are some multiple of root 2 but since they have no additional grid points, it is easier to see the size of the multipliers. For the square, there is no such confusion; the multipliers follow this sequence: 1, 1, 2, 2, 3, 3, 4, 4, 5,... | {{:spiralTri0.svg}} | 
^ ::: | ::: | {{:spiralS0.svg|400x300}} | 
^ tricky spirals | What are the sequences of multipliers for these spirals? || 
^ ::: |  {{:spiralHex0.svg|400x300}}  |  {{:spiralPent0.svg|400x300}}  | 
^ ::: |  {{:spiralHex2.svg|400x300}}  |  {{:spiralHex1.svg|400x300}}   | 

===== Spiral Connections ===== 
{{ :spirals0.svg?400x300|square spiral}}
One tool to understand patterns is to construct a table that breaks up the information. Below I have constructed a table with the length of each segment starting from the center of the spiral. As well, a running total length is calculated. 
^ length per side ^ total length |
|  1  |  **1**  |
|  1  |  2  |
|  2  |  **4**  |
|  2  |  6  |
|  3  |  **9**  |
|  3  |  12  |
|  4  |  **16**  |
|  4  |  20  |
|  5  |  **25**  |
|  5  |  30  |
|  6  |  **36**  |
|  6  |  42  |
|  7  |  **49**  |
|  7  |  56  |
|  8  |  **64**  |
|  8  |  72  |
|  9  |  **81**  |

A pattern in the total length can be seen. There are a number of ways to think about why this arises. For instance, each unit line segment in the spiral can be associated with a single dot in the grid which can in turn be associated with the square up and to its left. Total length becomes equivalent to area of a rectangle (which half the time is a square). 

In a related way, the sequence of total lengths can be shown zig-zagging down the times table. Since the sequence of square numbers are so useful, it's probably a good idea to highlight them. In this sequence, you may recognize that each square can be seen as a sum of consecutive odd numbers.

^  X  ^  1  ^  2  ^  3  ^  4  ^  5  ^  6  ^  7  ^  8  ^  9  |
^  1  ^  //**1**//  |  //**2**//  |  3  |  4  |  5  |  6  |  7  |  8  |  9  |
^  2  |  2  ^  //**4**//  |  //**6**//  |  8  |  10  |  12  |  14  |  16  |  18  |
^  3  |  3  |  6  ^  //**9**//  |  //**12**//  |  15  |  18  |  21  |  24  |  27  |
^  4  |  4  |  8  |  12  ^  //**16**//  |  //**20**//  |  24  |  28  |  32  |  36  |
^  5  |  5  |  10  |  15  |  20  ^  //**25**//  |  //**30**//  |  35  |  40  |  45  |
^  6  |  6  |  12  |  18  |  24  |  30  ^  //**36**//  |  //**42**//  |  48  |  54  |
^  7  |  7  |  14  |  21  |  28  |  35  |  42  ^  //**49**//  |  //**56**//  |  63  |
^  8  |  8  |  16  |  24  |  32  |  40  |  48  |  56  ^  //**64**//  |  //**72**//  |
^  9  |  9  |  18  |  27  |  36  |  45  |  54  |  63  |  72  ^  //**81**//  |

==== stepping through a triangular spiral ====
{{ :spiraltri0.svg?400x300|triangular spiral}}
The triangular spiral to the right needs a little twist to better understand its pattern. Previously, the idea of multipliers was discussed as a means of talking about the lengths of line segments. Here, I would like to replace the term multiplier by using the idea of //step//s. To describe the original triangle we would say, 3 steps: one DR (Down & Right), one L (Left), one UR (Up&Right). Even though the steps are different lengths, we will consider them the same unit. This imprecision is to overcome the problem of not using a triangular grid.

^ # steps per side ^ total # steps |
|  1  |  1  |
|  2  |  3  |
|  3  |  6  |
|  4  |  10  |
|  5  |  15  |
|  6  |  21  |
|  7  |  28  |
|  8  |  36  |
|  9  |  45  |

Here the path the sequence of total # steps has a bit of a jumping zigzag.
^  X  ^  1  ^  2  ^  3  ^  4  ^  5  ^  6  ^  7  ^  8  ^  9  |
^  1  |  //**1**//  |  2  |  //**3**//  |  4  |  5  |  6  |  7  |  8  |  9  |
^  2  |  2  |  4  |  //**6**//  |  8  |  //**10**//  |  12  |  14  |  16  |  18  |
^  3  |  3  |  6  |  9  |  12  |  //**15**//  |  18  |  //**21**//  |  24  |  27  |
^  4  |  4  |  8  |  12  |  16  |  20  |  24  |  //**28**//  |  32  |  //**36**//  |
^  5  |  5  |  10  |  15  |  20  |  25  |  30  |  35  |  40  |  //**45**//  |
^  6  |  6  |  12  |  18  |  24  |  30  |  36  |  42  |  48  |  54  |
^  7  |  7  |  14  |  21  |  28  |  35  |  42  |  49  |  56  |  63  |
^  8  |  8  |  16  |  24  |  32  |  40  |  48  |  56  |  64  |  72  |
^  9  |  9  |  18  |  27  |  36  |  45  |  54  |  63  |  72  |  81  |