How to Create a Spirallic Spin Particle Path or Necklace Form or Spherical Border

The Tutorial

1. 
   
   Open a new Excel workbook. Have ready 2 newly named worksheets in the workbook, one for the DATA, the other for the CHART. Title them so please -- it makes formulas easier to understand later.
2. Set Preferences

   1. In Preferences General, set Use R1C1 checkbox to off.
   2. Under View, set show row and column headings to On or checked and show gridlines. Show scroll bars and sheet tabs and outline symbols and zero values. Show formula bar by default and show status bar. It is very important to show the formula bar.

3. Create some Named Variables.

   1. Into cell A1, type AjRows.
   2. Into cell B1, type GM (for Golden Mean).
   3. Into cell C1, type Factor1.
   4. Into D1, type KEY.
   5. Into cell E1, type Number. 
4. Select cells A1:E2 and Insert Name Create in Top Row. Into cell A2, input 2880 (as the number of adjusted rows in the data-set). In cell B2, input the formula "=(sqrt(5)-1)/2" (without quote marks). This is the Golden Mean or Ratio or Proportion, known since Euclid's time, such that a:b as b:(a+b), and is quadratic and has many special properties, such as phyllotaxis in Nature -- it's used because it maintains in proportion the square of a number to the number such that for the Pythagorean Theorem, a given Pythagorean triplet will grow in constant proportion to itself, and since the sine and cosine functions are Pythagorean functions, it applies to them as well.
5. In cell C2, input .125 or "=1/8" (w/o quotes). Into cell D2, input "=36" (w/o quotes). Input 1 into cell E1.
6. Create some more Named Variables.

   1. Into cell A3, type Tip.
   2. Into cell B3, type Base.
   3. Into cell C3, type Spheroids.
   4. Into cell D3, type ShrinkExpand.
   5. Into cell E3, type PiDivisor.
   6. Into cell G3, type Lucky.
   7. Select cells A3:G4 and Insert Name Create in Top Row.
   8. Into cell A4, input "=B4*12*PI()" -- the empty parentheses for pi are correct. Do Insert Name Define name as Tip for cell A4.
   9. Into B4, input 1712, which is 2^4 * 107 (where 2^4 means 2 raised to the power of 4).
   10. Into cell C4, type 32 for now -- the lookup table that is going to be prepared is set to take any number of spheroids up to 64. (Actually 100 but the detail gets hard to see.)
   11. Into cell D4, type 1 for now. Setting up another plot column set of data with ShrinkExpand set to a different value will insert or surround one circle of spheroids with another one.
   12. Into cell E4, input 180. Differentiating this variable creates special effects.
   13. Into cell G4, input the Lucky number 63.
   14. Select cells A2:G2 and Format Cells Border  thick line Top Bottom Sides Center and format number with 4 decimal places. Do the same for cell range A4:G4.
7. Put the entire sheet in font Lucida Fax size 9 by selecting the very top left cell between A and 1 for column and row and thus select the entire worksheet. Generally, viewing 4 decimal places will inform one of the differences in the sine and cosine function results.
8. Create the Column Headings.

   1. Into cell A5, type Base t.
   2. Into cell B5, type c_.
   3. Into cell C5, type Cos (for cosine).
   4. Into cell D5, type Sin (for sine).
   5. Into cell E5, type Main X.
   6. Into cell F5, type Main Y.
   7. Base t is the number of turns and also the total distance in terms of points a particle would travel past. As you will soon see, the total variation in the column values is from +64,540.8795 to -64,540.8795, or +129,081.7590 absolutely. This is over 2881 rows. +129,081.7590/2881= 44.8045 valuation units per chart point. Each decrement is about -44.8201 and 1/44.8201 = .0223 … so the points are very very close together and the accuracy of the graph is very good. Charting 32 spheres over about 2880 rows, there are 2880/32 = 90 points per sphere or spheroid. They are termed spheroids because they are not perfect spheres. But they are pretty close -- how exactly square your chart area is matters quite a bit as well and that can be difficult to get right, as Excel has not provided a parameter for setting it precisely. The c_ in cell B5 stands for constant; you will see that the formulation varies the value of the constant with the number of spheroids the user inputs.
9. Input the t and c_ Formulas.

   1. Into cell B6, input the formula, "=Base". Do Edit Go To cell range B6:B2886 and do Edit Fill Down.
   2. Into cell A6, input "=If(odd(Spheroids)=Spheroids,0,Tip)". This formula states that if the number of spheroids input is odd, result=0, else result=Tip (where Tip was defined above as 12*PI()*B4 or 1712). In the case of the result becoming 0, the column will decrement to twice Tip, negatively from 0. The number of spheroids charted will be odd, eg. 31.
   3. Do Edit Go To cell range A7:A2886 and input "=((A6+(-Tip*2)/(AjRows)))" into cell A7, then do Edit Fill Down. The value in cell A2886 should be decremented to -64,540.8795 if spheroids is even (eg. at 32), else it will equal 2*-64,540.8795. Please see Note 6 in the Tips section regarding differentiating cells A7 and A6 because their formulas are different
10. (Adjusted) Cosine and Sine formulas.

    1. Do Edit Go To cell range C6:C2886 and input into cell C6 the formula, "=Spheroids/KEY*(cos((ROW()-6)*Number*PI()/PiDivisor*Factor1))" (without the quotation marks, as always). Do Edit Fill Down.
    2. The KEY number is an other means of warping the output; currently, it's set to 1, non-warping. All it is really doing is applying a fraction to the main cosine function. By taking the cosine of the cell 6 rows up from cell C6, we are taking the cosine of 0 to begin with, then in subsequent rows, take the cosine up to 360 and in cycles then to 2880+6. 2880/360=8. So there are 8 cycles. Number is a variable for getting partial circle effects like animation motion likenesses when a fraction because the next piece, *PI()/PiDivisor. converts from pi radians to degrees and vice versa. Recall that PiDivisor is set to 180 degrees. Multiplying by Factor1=.125 is taking back the 8 by 1/8th.
    3. Do Edit Go To cell range D6:D2886 and input into cell D6 the following formula: "=Spheroids/KEY*(sin((ROW()-6)*Number*PI()/PiDivisor*Factor1))".
    4. Do Edit Fill Down. This is the sine or y function in place of the x cosine function just done.
11. Input the Main X and Main Y Formulas. Do Edit Go To cell range E6:E2886 and input into cell E6 the following formula: "=((sin(A6/(B6*2))*GM*cos(A6)*GM*(cos(A6/(B6*2)))*GM)+C6)/ShrinkExpand". This is the heart of the Spherical Helix formula and is sin * cos * cos. Do Edit Fill Down.
12. Do Edit Go To cell range F6:F2886 and input the formula "=((sin(A6/(B6*2))*GM*sin(A6)*GM*(cos(A6/(B6*2)))*GM)+D6)/ShrinkExpand". This is sin * sin * cos. Be very careful to match your parentheses exactly as given.

    1. If you are getting an error, it's probably because of a missing parenthesis, so count that the lefts = the rights and look to see that you have them placed exactly as given. If you are getting an undefined name error, it means that "GM" is not properly associated with cell B2 -- go back and Define Name that variable again, without quotation marks. Otherwise, it's ShrinkExpand for cell D4. If either of those cells or any previous cell contains an error value, go back and fix it according to the above instructions.
13. While the sheet will now produce the chart wanted, for small numbers of spheroids like 1 or 2 or 3, they cannot appear in a ring properly. A solution for that contingency has been worked out.

    1. Do Edit Go To cell range I6:I69 and input 1 into cell I6 and then do Edit Fill Series Columns Linear, Day, no trend, Step Value 1 and blank stop value; press OK. That should input the series from 1 to 64 in that cell range.
    2. Do Edit Go To cell range J6:J69 and input into cell J6 .125 and then do Edit Fill Down. That will input the constant value of .125 in that cell range.
    3. Do Edit Go To cell range K6:K69 and input "=I6*$K$35/$I$35" and do Edit Fill Down and then input the value .125 in cell K35. Do Edit Go To cell range I6:K69 and Insert Name Define Name Looker.
    4. Go to Factor1 cell C2. Input the formula, "=Vlookup(Abs(Spheroids),Looker,2)". If that last number is changed from 3 to 2 in the VLookup Formula just given, it will always return .125, otherwise it will adjust for the number of spheroids input the value of Factor1 (so that will no longer be an available variable). Change the font of cell C2 to dark blue italic or something red or something that helps one remember not to change it. If your chart does not look right, try changing this formula to "=Vlookup(Abs(Spheroids),Looker,3)".
14. It is also a good idea to do Insert Comment a copy of all the formulas so far into the cells so there are always the original formulas should they ever be overwritten. Do that now for cell C2 until you see the red corner flag and remember to do it for all the other formulas later please. The Abs() Absolute function allows one to input a negative number of spheroids; the effect will be to flip the graph 180 degrees horizontally (from left to right or right to left), as one can tell with an odd number of spheroids.