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

Two Parts:The TutorialCreate the Charts

You will learn to create a very versatile curve, created in 1994 or so. Possible usages include Recycling Economics, or any cyclical activity that can be charted, as when resources are allocated and used evenly and equally by a group of chemicals, machines, humans, competitors or political groups that share power/energy, other resources during throughput portion of Input-Throughput-Output models (e.g., lightning fast group-think in reaction to a TV/media buzzword/ad), Quasi-physics application to particle paths undergoing spin within an orbital shell for example, and more obviously, as a neck wear design or bracelet, or spherical border for a stationery design that's truly unique (it is not theoretically modern to create a graphic for an electron's path anymore however). It's a metaphor for brainwaves transmitted through neuron groups / neural net when thought is repetitive, as is so common to creatures of habit.

You'll learn how to use Microsoft Excel in about ten columns and under 3000 rows to build from variables to data-sets with at most two sentence-long trig formulas that you can modify to make graphic artwork that also has a firm scientific and mathematical basis. (Like a cellphone, you do not need to know how all the math relates and works to use it.) This curve may help you make a great first impression.

Part 1
The Tutorial

  1. Image titled NW Apple Tree in Yard.png

    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. 2
    Set Preferences
    • In Preferences General, set Use R1C1 checkbox to off.
    • 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.
      Image titled OP Boats Beached.png
  3. 3
    Create some Named Variables.
    • Into cell A1, type AjRows.
    • Into cell B1, type GM (for Golden Mean).
    • Into cell C1, type Factor1.
    • Into D1, type KEY.
    • Into cell E1, type Number.
  4. 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. 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. 6
    Create some more Named Variables.
    • Into cell A3, type Tip.
    • Into cell B3, type Base.
    • Into cell C3, type Spheroids.
    • Into cell D3, type ShrinkExpand.
    • Into cell E3, type PiDivisor.
    • Into cell G3, type Lucky.
    • Select cells A3:G4 and Insert Name Create in Top Row.
    • Into cell A4, input "=B4*12*PI()" -- the empty parentheses for pi are correct. Do Insert Name Define name as Tip for cell A4.
    • Into B4, input 1712, which is 2^4 * 107 (where 2^4 means 2 raised to the power of 4).
    • 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.)
    • 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.
    • Into cell E4, input 180. Differentiating this variable creates special effects.
    • Into cell G4, input the Lucky number 63.
    • 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. 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. 8
    Create the Column Headings.
    • Into cell A5, type Base t.
    • Into cell B5, type c_.
    • Into cell C5, type Cos (for cosine).
    • Into cell D5, type Sin (for sine).
    • Into cell E5, type Main X.
    • Into cell F5, type Main Y.
    • 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. 9
    Input the t and c_ Formulas.
    • Into cell B6, input the formula, "=Base". Do Edit Go To cell range B6:B2886 and do Edit Fill Down.
    • 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.
    • 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. 10
    (Adjusted) Cosine and Sine formulas.
    • 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.
    • 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.
    • 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))".
    • Do Edit Fill Down. This is the sine or y function in place of the x cosine function just done.
  11. 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. 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.
    • 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. 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.
    • 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.
    • 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.
    • 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.
    • 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. 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.

Part 2
Create the Charts

  1. 1
    Have a blank worksheet ready to copy a new chart into and expand and format at will.
    • Do Edit Go To cell range E6:F2886. Press the Charts button on the Ribbon, All, or Chart Wizard or Insert Chart. Select chart type Scatter Smoothed Line Scatter. Command c copy it and access the new worksheet and command v paste it into the new worksheet. Your chart should look like the example above except for the bronze tinge would be white and the default line might be black and too thick or thin. Adjust the chart size by dragging the bottom right corner until you get a square plot area and a circular ring. Get rid of the Vertical and Horizontal Axes and gridlines via the Chart Layout tab. Clicking on the plot area will give access to the gradient style, etc. Double-clicking on the chart plot line itself will give access to changing that. A value of line thickness = 1 is recommended. Mission accomplished!! Shrink the chart on the original Data worksheet and put it at the bottom of the data. Split the window so that you can see the small chart and the last rows of data.
    • Optional: Create a Double "Rainbow" or a ring within a ring, like benzene rings and tightly packed animal cells such as a hornet's stinger. Select D3:D4 and do Edit Copy to H3:H4. Edit ShrinkExpand by adding a 2 to it to become ShrinkExpand2. Insert Name Create Top Row while cell range H3:H4 is still selected. Select cell E6. Select over the formula in the formula bar and command c copy it (do not copy the cell -- copy exactly only the formula itself in the formula bar). Do Edit Go To G6:G2886. Mouse-select in the formula bar and paste the formula just copied. Edit the last ShrinkExpand to make it ShrinkExpand2. Do Edit Fill Down. Select cell F6. Select over the formula in the formula bar and command c Copy it. Do not copy the cell and paste it into the new cell -- that won't work out right. Do Edit Go To cell range H6:H2886. Edit in the formula bar the last ShrinkExpand to become ShrinkExpand2. Press Enter or Return. Do Edit Fill Down. There should be all zeroes because the value in ShrinkExpand2 = 0.
    • Select cell G5 and type Second X and in cell H5 type Second Y. Now go to cell H4 and type in 1.5 and go to the original ShrinkExpand in cell D4 and input 2. On the Chart worksheet, select menuitem Chart Add Data and then select back on the Data sheet cells G6:H2886 and hit OK. Somehow it comes out wrong sometimes and one must click on the new chart series in the Chart worksheet and type in G's for E's in the plot series in the formula bar -- then it's OK.
    • There should now be two series: 1) =SERIES(,Data!$E$6:$E$2886,Data!$F$6:$F$2886,1) and 2) =SERIES(,Data!$G$6:$G$2886,Data!$H$6:$H$2886,2). They should be just touching with the first series inside the second series. Select a good color for the second series and line thickness should probably be set to 1 for each.
    • Well done!! Whether one also adds the 2nd series to their Data worksheet bottom chart is a matter of preference -- generally, it's cleaner to see effects take place on one series at a time quickly in miniature.
  2. 2
    Above is the chart for 32 spheroids.


  • Sine = y/hypotenuse or radius r, usually set to 1, so Sin(n) where n are degrees of arc in the circle, is the y axis distance upwards or downwards of the {x, y} coordinate in the Cartesian Plane. Cosine = x/h or x/r where hypotenuse h or radius r again = 1 in the Unit Circle, so Cos(n) = the degrees of arc as measured left or right along the x axis horizontally. Together, {Cos(30*PI()/180), Sin(30*PI()/180)} successfully converts from radians to degrees and both give you the {x, y} coordinate of the circle data point at precisely 30 degrees, measured upwards from 0 degrees at the right of {0, 0}, 1 unit away. There are 360 degrees of arc in a circle, or 720 twice around. If I want to place a "natural" 8 spheroids in a ring along a circle, I must have 2880 (8*360) paired coordinates, times .125 within the formula, as .125 is 1/8th of something. Thus, I put in the formula for a helix in 2880 * 2 columns, and add to it the basic ring of Cos and Sin, taken of the row the formula is in, from say Row 2 to Row 2882, to get my helices to proceed around in a ring. Thus, the formula for the cosine looks like this: Cos((Row()-2)*PI()/180*.125) in cell say A2, and that will give me the Cosine of zero degrees. It then proceeds to increment automatically when I fill the formula down all the way to cell A2882, taking next the cosine of 1/8th of 1 degree, then 1/8th of 2 degrees in the next row, etc. This way, at 2882-2 = 2880, 2880/8 = 360, and I have described one full circle with the data points needed to describe 8 spheroids within those 360 degrees. Because when the helix or spheroid calculation occurs a column or two over to the right, it thinks there are 8 circles, since I am not multiplying in that formula by .125, OK? But what I am actually not doing is overlaying 8 helical spheres atop each other and then shifting them so they'll adjoin around a 360 degree circle. Instead, I am forming the curve out of one line, one spiralling line. That is my main advance, and it is called The Garthwaite Curve, as it appears in no text on Standard Curves or anywhere online that I could find in the course of 4 and now many more years of research. And until one knows some other magical numbers that play into the spreadsheet, mostly what one gets is sheer chaos, like I did for 4 years. But I could see enough of a concise curve at times, just hints of it, to keep at it. I finally found the numerical relation! It occurs just once in every 113,000 triplets of numbers and is what's called a "Standing Wave" in Fractal Math. It is a very special kind of Order, or Hyper-Order. Since I found out how to make 8, I went on to make 24, then 30, then 6, 1, then any number up to 100 spheroids in a circle. Took me another year and a half to do so. The curve may very well be important to Recycling Economics, Finance, and a host of other cyclical and spirallic forms of endeavor, with periods of predictable growth and decline, based on well-known events, both natural and artificial. I believe it's an important curve. In the first image above, you can see one nestled inside of another, like electron orbital rings, with spin part of their path.
  • Try setting the number of spheroids as the larger birthdate divided by the smaller one, or the average of the two, or their difference, or inverse quotient -- be creative! A number like "=(1954/9/2)" may be rounded to zero decimal places by entering it as "=round(1954/9/2,0)".
  • The normalizer of 36 may certainly be self-adjusted and filled down. 30 might be better. If you're experiencing difficulties and cannot recall why your chart looks so radically different, try resetting the last number in the VLookup Formula in cell C2 back to 2 or change it to 3 -- it refers to which column, the second or third, it picks up the value corresponding to the value of spheroids it looked up in column I from. If your chart does not look spherical, switch from 2 to 3 or vice versa.
  • There are thousands of variations on this basic model that are quite exciting. Please conduct a search on FieryTrig in Google to follow some of my further ideas and theories. For example, think of each spheroid as a leaf on a plant that is centered at {0,0} and where the leaves originate at {0,0), emanate, get blown gently by a passing breeze of an overall data increment in a U-shaped form, and each leaf is returning to zero so that they are extended rays or petals. Such can be accomplished with Neutral Operations …. and that's another article and blog … already written! Enjoy!
  • Please, for clarification and validation of the 4-5 year process of trial and error to find these well-ordered curves, see my website* and let the images of the many failed attempts inform you of those 5 years. The odds of finding the right combination of numbers were once calculated to be 113,000 to 1 and the trials did indeed run to the tens of thousands. There wasn't any guiding literature or text stating the relationship was based on 12pi instead of 10pi, and it's not obvious why that is.
  • It is again a good idea to do Insert Comment a copy of all the formulas into each of the critical top cells so one always has the original formulas should one ever overwrite them. Cell A7's formula is also different than that of A6, so remember to Insert Comment for that one too and change the font or background color or in some way make it distinct from A6.
  • At this point, omitted are the birth dates from the sine and cosine formulas, or the constant in B4, or column A, to keep things simple. To use the birth dates, (so long as an Insert Comment was done to preserve the original formula) input the following formula into cell B4: "=NewDate2" (without the quote marks). And into cell A4 (so long as an Insert Comment was done to preserve the original formula), input the following formula: "=(NewDate1+NewDate2+Lucky)" A4 should have the result 210 in it. This is the new value of the variable Tip. Cell B4 should contain the value 38 and so should the column beneath it from B6:B2886. There should result a design resembling a tokamak. Other birth dates and Lucky numbers will result in different designs. One may also change cells B6 TO B9 to the following formulas: in cell B6, input the formula "=0"; in cell B7 input the formula, "=NewDate2/NewDate1"; into cell B8 input the formula, "=If(NewDate1>NewDate2,NewDate1,NewDate2)"; and into cell B9 input either the formula "=NewDate2/NewDate1" or "=NewDate1/NewDate2". Do Edit Go To cell range B10:B2886 and with cell B10 the active hi-lighted cell, input the formula, "=B6", and then do Edit Fill Down. This will create a leaf pattern that returns to 0, or a point of emanation, prior to extending itself outwards. It might be a good idea to set AjRows to 360 and Spheroids to 1, keeping the design more compact, less complex. Please, also, if changing AjRows, down to 360, adjust the Chart Series down to 360 as well: of the two series, Series1 should read "=SERIES(,Data!$E$6:$E$366,Data!$F$6:$F$366,1)" and Series2, "=SERIES(,Data!$G$6:$G$366,Data!$H$6:$H$366,2)" via directly double-clicking on each plot-series and editing their formula up in the formula bar on the Chart worksheet. It would be a good idea to edit the mini-chart too at the bottom of the Data worksheet (it may be necessary to drag-expand it a bit first).
  • It is true that the number 210 was stumbled upon by accident and may work without multiplying by 12pi. This is because if one uses 210 and the total decrement is -420, 420/360 is 7/6 absolutely, which is a good relation numerically to the number of spheres (32 = 2^5 and 36 works well too at 2^2 * 3^2) and to 12 of 12 pi, if that is utilized. So choose numbers that factor "well" (in interesting ways) with 360. Off to the side, one can sum the NewDates and Lucky number and divide by 360 to get an idea if the number will work well. The factors of 360 are 2^3 * 3^2 * 5, which is a lot of variable factors when you combine the numbers in all possible ways. It can be figured out by multiplying the factor's exponents+1 together, so it's (3+1)*(2+1)*(1+1) = 24 factors. 360 has 24 factors 1, 2, 3, 4, 5, 6, 8, 9, 10, 12, 15, 18, 20, 24, 30, 36, 40, 45, 60, 72, 90, 120, 180, 360. 420 is 2^2 * 3 * 5 * 7 so it has (2+1)*(1+1)*(1+1)*(1+1) = 24 factors also, a good number of which it shares in common with 360.
  • To factor a number, find a small work area off to the upper right with two columns and at least 20 rows. Suppose you choose J4:K24. In the uppermost right cell of K4, input the number you want to factor. In the cell range K5:K24 below it, enter the formula "=K4/J5" and do Edit Fill Down. There will be errors -- ignore them. Let's say the number to factor is 720. Start out by entering 2 into cell J5. 360 will appear in cell K5. Enter another 2 in cell J6 and 180 will appear in cell K6. Enter another 2 in cell J7 and 90 will appear in K7. Enter another 2, because 90 is divisible by 2, in cell J8 and 45 will appear in cell K8. Enter 3 in cell J9 and 15 will appear in K9. Enter 3 in cell J10 and 5 will appear in cell K10. Enter 5 in cell J11, and 1 will appear in cell K11, and now one has all the least factors of 720, i.e. 2^4 * 3^2 * 5. There are thus (4+1)*(2+1)*(1+1) = 30 factors of 720 in all. One can figure those out from by adding 1 to the exponents, per term.
  • Please also see How to Create a Powerful Trigonometric Design in Excel for further instructions on the tokamak design.
  • Doing lots of changes? Saves them on a new SAVES worksheet and grab the upper 10 rows and columns and Copy and Paste them once as is, and then once again under a Paste Special Values, with the latest chart underneath them; from there one can play with special Picture effects, etc. off to the right of the chart, leaving notes as one goes of effects one likes and learns from.
  • Ready for some advanced geometrical effects? First, follow what was done to get this far and produce chaotic effects by random relationships between A4 and B4. The basic formula for a spherical helix is: "1) x = sin[t/(2c)]cos t; 2) y = sin[t/(2c)] sin t; and 3) z = cos [t/(2c)] where 1. c=5.0; 0<t<10π" per "CRC Standard Curves and Surfaces" by David von Seggern, CRC Press, Boca Raton, FL, 1993, ISBN 0-8493-0196-3, pg. 264; therefore, if the average of t is 5π, then (t/2c) = (5π/10) or (π/2); and thus if the maximum attained by t is 10π, then (t/2c)=(10π/10)=π; ergo sin(π)=0, cos(π)=1, sin(0)=0, cos(0)=1. sin(π/2)=1 and cos(π/2)=0 (working from the ends to the middle). Combining z into x and y gives x = sin t/2c cos t cos t/2c and y = sin t/2c sin t cos t/2c. Find formulas for curves and surfaces by googling or buying the CRC book referred to above (the latter's recommended as a good starting point). This formula leads to trig(π^3) in a way. Except 12π is being used, not 10π. Try 10π though and see the results: it does not produce the proper nodes. However, as an example from the book, one can produce a single helix like a DNA Helix when "1) x = a cos(t + 2πi/n) for i = 1,...,n; 2) Y = a sin(t + 2πi/n) for i = 1,...,n; 3) z = t/(2πc) where 1. a=.3, c=3.0, n=2; 0<t<6π" and that gives 6 loops to the helix. To create the second offsetting erect helix, probably start at x=-1, y=0 (look at your previous output to see where that is if it's hard to think through).
  • The relationship between A4 and B4 and the least value of B4 for good spheres took 5 years to find. That is why this is "The Garthwaite Curve" -- Standard texts on curves were searched and it was found it nowhere. It was derived from David von Seggern's "CRC Standard Curves and Surfaces" formula for the Spherical Helix [7.1.4 page 264], but the idea of setting the spheroids into a ring was personal and took many hours of trial and error to get right, as it occurs only once in about every 113,000 values, though a lot of the chaos found approached it in pieces -- working until any number of spheroids would appear in a circle took quite a bit of patience and concentration.
  • There is now made an animation showing a small red curve tracing along as it moves within the larger external Garthwaite Curve as seen below and cited in the "Sources and Citations" section at the bottom of this page which involves two macros and some defined name areas on the Data sheet additionally. If interested in pursuing this, please contact me at on my Talk Page and further instructions will be supplied. GW Curve Non-flash Movie
  • Designed by a successful ex-CFO who has passed the CPA Exams, this curve is one long spiral, not one spheroid rotated and pasted beside the next, as it follows a cosine and sine circle pathway for its spheroids to occupy. It is the base curve for many of the other articles you will find by this author-editor, including chaotic forms where the processing by each spheroid does not conform to a norm. You can find out more about how it works by reading the article, How to Compare Two Methods of Creating a Spherical Helix, or online under spherical helix. It is highly ordered and highly functional.
  • Spirals based on the number Phi are probably how nearly closed-off tubes and organs begin to grow, or terminate at a point. Imagine learning about the anatomy of an animal by taking wafer-thin slices, 1 cell thick, and analyzing that group of curves ... at most resolutions. The spheroids may be joined at points or by walls, and the entire curve's ends meet with accuracy of 10^-13 if so desired.
  • Overlapping spheroids have also been achieved, as well as short strands which do not complete a full cycle/circle. The line thickness setting determines whether it looks like it has a surface or not -- this is a prime structure.

Helpful Guidance

  1. Make use of helper articles when proceeding through this tutorial:
    • See the article How to Create a Sin and Cos Circle in Excel for help with understanding how to create a circle with trigonometry and for a list of articles related to Excel, Geometric and/or Trigonometric Art, Charting/Diagramming and Algebraic Formulation.
    • For more art charts and graphs, you might also want to click on Category:Microsoft Excel Imagery, Category:Mathematics, Category:Spreadsheets or Category:Graphics to view many Excel worksheets and charts where Trigonometry, Geometry and Calculus have been turned into Art, or simply click on the category as appears in the upper right white portion of this page, or at the bottom left of the page.
    • Here is a list of most of the articles using or related to this curve:

How to Create a Powerful Trigonometric Design in Excel, How to Make a Square of Spherical Helixes, How to Create a Slideshow of Excel Images, How to Chart Orderly Chaos, How to Create a Cyclical Chart Using Spheroids, How to Create Spheroidal Asymptotes and Skewed Sphere Ring, How to Create a Tekeporter Frame Image in Excel, How to Create a Pink Love Note of Spheres in Form of a Heart, How to Create Floral and Other Images with Trig and Neutral Operations, How to Create a Lemniscate Spheroid Curve, How to Create Artistic Patterns in Microsoft Excel, How to Create the One Sphere Pattern in Microsoft Excel, How to Create a Line of Spheres Pattern in Microsoft Excel, How to Create a Necklace Pattern in Microsoft Excel, How to Create an S Curve Pattern in Microsoft Excel, How to Create a Different Necklace Pattern in Microsoft Excel, How to Program Excel to Show Spheroids Visiting Their Home Planet, How to Create a Dakini and Boddhisattva Aspect of the Mother Planet, How to Create the Photon Emission Image, How to Create the 3 Transformative Mother Planet Receptacle Images, How to Create the Idea of an Idea Image, How to Acquire the Black Mosaic Tile Image via Excel, How to Acquire a Conical Helix with Spheroids Image in Excel, How to Compare Two Methods of Creating a Spherical Helix, How to Solve Random Beads, Overlapping Spiral, Asymptotic Axes Problems, How to Approximate Arc Length Using the Distance Formula, How to Make Your Excel Curve Solid or Transparent, How to Create a Sin and Cos Circle in Excel, How to Acquire and Logically Perceive the Rose Curve


  • Follow each step. Missing a step would lead to a critical error.
  • It would also be a good idea to Insert Comment for the original variable values as one may want to play with those and finding them in all these instructions would be more difficult than simply reading an on-hand Comment.

Sources and Citations

  • The Garthwaite Curve shown was first created by Chris Garthwaite in May of 1994 and is based upon the Spherical Helix found on page 264 of "CRC Standard Curves and Surfaces" by David von Seggern, CRC, ANN ARBOR, 1993. ISBN 0-8493-0196-3
  • The workbook used for this article was " THE Garthwaite Curve" (with spaces).
Show more... (2)

Article Info

Categories: Graphics | Microsoft Excel Imagery