How to Create Floral and Other Images with Trig and Neutral Operations

Three Parts:The TutorialExplanatory Charts, Diagrams, PhotosHelpful Guidance

Excel has many capabilities, one of which is to create some wonderful floral and other imagery via trig and other math genres, Learn to make the 'chrysanthemum' (or lotus) below in relatively few steps.

Part 1
The Tutorial

  1. 1
    Open a new Excel workbook and create 3 worksheets: C.Data, Chrys Chart and Saves. Save the workbook into a logical file folder. If you tap on the green font "Ticklezzz" below, there will appear an animation of the curve to be created; hit the browser's Back button to return from YouTube here. It was done with random number generation at the tips and pi approximation to 15 digits for a constant recalc, and recorded in QuickTime Player.

  2. 2
    Set Preferences: Open Preferences in the Excel menu. Recommended Settings: Set General to R1C1 Off and Show the 10 Most Recent Documents; Edit - set all the Top options to checked except Automatically Convert Date System. Display number of decimal places = blank (for integers preferred), Preserve display of dates and set 30 for 21st century cutoff; View - show Formula Bar and Status Bar, hover for comments and all of Objects, Show gridlines and all boxes below that auto or checked; Chart - show chart names and data markers on hover. Leave rest unchecked for now; Calculation Automatic and calculate before save, Iteration limited, 100 (200 for the animation tho), max change .000 000 000 000 01 w/o spaces if you do goal-seeking a lot (like for the animation) and save external link values and use 1904 system; Error checking - check all; Save - save preview picture with new files and Save Autorecover after 5 minutes; Ribbon -- all checked except Hide group titles and Developer.
  3. 3
    Select the entire worksheet by selecting between the 1 of row 1 and the A of column A in the upper and left most corner of the C.Data worksheet. Do Format Cells Font Size 9 or 10 if you're comfortable with that and Format Number Number Decimal Places 4.
  4. 4
    Enter the Defined Name Variables and headers of row 1. A1: Enter 1, and Format Cell Number Number Custom "Degrees="00.00 w/ commas and Insert Name Define name Degrees for cell $A$1 and do Insert Comment and edit in "A1: Original Value is 1. Set to whatever pleases your aesthetic -- chaos is possible herewith tho! Low values work better. Notice that inputting a 2 gets rid of all those wonderful little Errors in the data!? Error which should probably rather be kept, as will be explained."; B1: Enter the column heading Neutrality Formulas; C1: Enter in the Defined Variable Name Converter; D1: Enter w/o quotes the formula "=(PI()^2)/((PI()^2)-(205))" and Insert Name Create in Left Row name Converter for cell $D$1 and Insert New Comment and edit in "D1: The original value is formula (to be entered in the formula bar as copied from here) of: =(PI()/180)^-0.1414 …. You could try 16 or 2 or-.01 or =PI()/2.5. Now=(PI()^2)/((PI()^2)-(205))"; E1: Enter the Defined Variable Name CFactor. F1: Enter 95 and Insert Name Create Left Row Name CFactor for cell $F$1 and do insert New Comment and edit in "F1: The original value is -5. Try entering 0 or 1 or 6 or =-PI() or =-EXP(1) (a fave) .. Which is the transcendental number e in XL .. Or 18 or any formula, preceded by an "=", and logically valid to Excel. Input of .00001 may result in something like a simple circle, if var(iable) Degrees doesn't equal 1 maybe? Currently= 95."
  5. 5
    Enter the column headings. A2: a; B2: b; C2: sin a; D2: cos a; E2: Factor sin a; F2: cos b.
  6. 6
    Enter the formulas. In cell A3, enter w/o quotes the formula "=Degrees" and Insert Name Define name Top for cell $A$3 and do Insert New Comment and edit in "A3: This cell has the Insert -Define(d) Name of "Top", as used by cells below it in their formulas. Its own reference is to the "Degrees" variable."; Edit Go To cell range A4:A363 and with A4 the active cell enter w/o quotes the formula "=A3-((2*Top)/360)" and Edit Fill Down and select cell A4 and Insert New Comment and edit in "A4: The effect of this formula, "=A3-((2*Top)/360)" is to decrement a series automatically through +Top to -Top (cell A3=Top), eg. When Top=1, the result in cell A363 at the bottom will be -1, the series having decremented thru 0 (in cell A183). If instead Degrees=360, then Top=360= cell A3's new value in the formula. A3=360. So 360-((2*360)/360)= (360-(720/360)) = 360-(2) = 358. Cell A362 near the bottom = -358 and cell A183 still = 0. The decrementing is uniform. If Degrees=Top= (-1), then the formula increments up to A363=+1 uniformly.".
  7. 7
    Into cell B3 enter w/o quotes the formula "=IF(ISERR(A3/(A3-1)),0,A3/(A3-1))" and do Insert New Comment and edit in "B3: The formula in this cell, "=A3/(A3-1)" is a Neutral Operation in that when Y= X/(X-1), then X + Y = X * Y, as does this cell =A3/(A3-1), similarly. But the cell-formula below is different!"; B4: Into cell B4 enter w/o quotes the formula "=-A4/(1-(1/A4))" and do Insert New Comment and edit in "B4: This cell's formula, "=-A4/(1-(1/A4))" is also a Neutral Operation -- that of a-b = a/b. So the effect is that there are rotating formulas thru various Neutralities." B5: Enter in w/o quotes the formula "=A5/(A5+1)" and insert New Comment and edit in "B5: This cell's formula, "=A5/(A5+1)" is another Neutrality., of a-b = a*b." B6: Enter into the cell w/o quotes the formula, "=A6/(1-(1/A6))" and Insert New Comment and edit in "B6: This cell's formula, "=A6/(1-(1/A6))" is another different Neutrality of a-b = a/b. Edit Go To cell range B7:B363 and Edit Fill Down, which will repeat the 4-sequence formulation above to the bottom.
  8. 8
    Edit Go To cell range C3:C363 and with C3 the active cell, enter the formula w/o quotes "=(SIN(CFactor*A3/(2*B3)*Converter)*COS(A3*CFactor*Converter))+COS((ROW()-3)*PI()/180)" and Edit Fill Down and Errors are on purpose and Do Insert New Comment and edit in "C3: This formula, =(SIN(CFactor*A3/(2*B3)*Converter)*COS(A3*CFactor*Converter)) +COS((ROW()-3)*PI()/180) looks a lot more wicked than it really is. The ±Top and Decrement in cell-column A, e.g. In A3, is divided by twice the Neutrality-result in B3, so that's "A3/(2*B3)". That result is multiplied by both CFactor, say = -5, and Converter, say = 1.7725501863 or =(PI()/180)^-0.1414. Then the SIN() function is applied to that result. The SIN() function returns a value from 0 to 1, having to do with a point on a curve, say a circle or an ellipse or a flower's edge. The above result is multiplied by a COS(ine) function performing the same way. So it's multiplying together 2 parts of the same angle. If sine is y/r and cos is x/r than it's achieved XY/r^2, right? Then, by adding to the above total product the LAST formula term, "+COS((ROW()-3)*PI()/180)", it's setting the result in a path around a circle (or cycle). For cell C3 in Row 3, the ROW() function returns the row the formula is in, row 3, and subtracts 3 from it; 3-3=0, so I'll get the COS of 0 which is 1. The "*PI()/180" part of the formula is Excel's way of converting sine formulas from angles to pi-radians and vice versa. Performed over 360 or 361 rows, the full circle is completed by the last term, as modified in form by the previous XY/r^2 relation ... that's it in a nutshell. Why's it called "sin(a)"? To fool intruders? No, that's just how the sheet formulas evolved, so it helps keep track of the Form-Evolution-Path."
  9. 9
    Edit Go To cell range D3:D363 and with D3 the active cell, enter the formula w/o quotes "=(SIN(CFactor*A3/(2*B3)*Converter)*SIN(A3*CFactor*Converter))+SIN((ROW()-3)*PI()/180)" and Edit Fill Down and the errors are on purpose and do Insert New Comment and edit in "D3: This formula works pretty much like the one in D3, except that it's using 3 SIN() functions together instead of SIN*COS+COS of cell C3. The formula, "=(SIN(CFactor*A3/(2*B3)*Converter)*SIN(A3*CFactor*Converter))+SIN((ROW()-3)*PI()/180)", is different in 2 respects: 1) the 2nd term, "SIN(A3*CFactor*Converter)", for sine = distance Y over radius r, i.e. y/r, is fully squared, so I get Y^2/r^2, instead of XY/r^2 as explained in cell C3; 2) the last term "SIN((ROW()-3)*PI()/180)", works with and against the final COS() function in cell C3 to complete the points around the circle, since elements of set {X=COS(), Y=SIN()} give the coordinates of circle points on the circumference (unless and until otherwise modified, as done here by the other terms), relative to a constant radius r. But the whole point is to modify the constancy of radius r to get the fun design! To understand this better, see the explanation of the formula in cell C3's comment please, or ask."
  10. 10
    That is all that's needed to produce the chart shown above, There are two more columns, however, for adventurous types. These next two steps may be skipped therefore, but they're easy. Into E1, enter "Sin(C3)" and Edit Fill Down to E3:E363. Do Insert New Comment and edit in "E3: The new formula, =SIN(1/CFactor*C3*(1/Converter)), is an attempt to see a Side View -- see Chart at bottom in blue and white. This formula, "=CFactor*C3" merely re-applies whatever's the value or formula-value-variable in CFactor in cell F1 to the formula in C3 … from the outside, as it were. Whereas earlier, Cfactor gets used internally in cell C3's formula. The column title, "Factor sin a" tells the story of what I'm doing. Currently the formula is set to =SIN(C3)" instead.
  11. 11
    Edit Go To cell range F3:F3633 and with F3 the active cell, enter w/o quotes the formula "=COS(D3)" and Edit Fill Down. Charting cell range E3:F363 I leave as an exercise to do. Perhaps Cos and Sin should be switched.

Part 2
Explanatory Charts, Diagrams, Photos

  • (dependent upon the tutorial data above)
  1. 1
    Create the Chart. Edit Go To cell range C3:D363 and use either Chart Wizard to create a new chart page or use the Ribbon and select Charts, All, scroll down to Scatter, Smoothed Line Scatter and OK and Cut or Copy the chart to the Chrys Chart worksheet's A1 cell. Hover over the lowest right corner until the double headed arrow appears and pull the chart open into a large square, Click on the Plot Background and choose Radial Fill dark Green right and Black left radial from Center, Double click on the series plot line and change the Line weight to 1 and color white. In Chart Layout, get rid of the axes and chart label, etc. unless you want to leave formulation notes and that is done via Chart Title Text Box options for MS Excel 2011 for the Mac. You are Done! The above jpg shows how it should have turned out

Part 3
Helpful Guidance

  1. 1
    Make use of helper articles when proceeding through this tutorial:
    • See the Related wikiHows below and the article How to Do the Sub Steps of Neutral Operations for a list of articles related to Excel, Geometric and/or Trigonometric Art, Charting/Diagramming and Algebraic Formulation relating to Neutral Operations.
    • For more art charts and graphs, you might also want to click on Category:Algebra, 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.


  • There are many variations possible with this simple data set of values, variable, formulas, etc. Here are just a few of the charts created from it, or with minor additions:
    Chrys In Wind.
    Neutrality Recursions
    Looped Asymptotes
    5 queens


  • Errors: This worksheet is special. Errors are desirable. They evaluate to 0 and return the branches of the form back to the Origin. There should be #DIV/0! Errors in cells C3 and D3, C7 and D7, (and in columns E and F as well), and in every 4th cell thereafter; that is normal for this worksheet and chart. A picture of the correct data returns is immediately below.
  • One may select in the Chart Menu Select Source Data and it will bring up a window where one may opt to have empty cells show as gaps or zero or be connected data points with a line otherwise, and there's also an option there to show data in hidden rows and columns -- this business of displaying error values as zeroes is unrelated, and not all error values chart as zeroes is my recollection (see ISNA() in Excel Help for more detail).
  • If you have errors or error values, either the sheet in incomplete and needs further input or Lookup Tables for critical variables, or perhaps you've made a mistake somewhere along the line. If the instructions have been completed and there are still undesirable errors because the Chart is Not Right, select the cell that has the error value that is furthest left and topmost first. Look for a typo in a formula or unmatched parentheses. Possibly, a Defined Name is wrong -- they need to be input into the formulas exactly as they were defined. Do Insert Name Define to check. If you have DIV/0!, I do not, so look for a variable that somehow did not get filled in with a value perhaps. At any rate, what you want to do is select the cell with the error, and after checking all those typical errors, do Tools Auditing Trace Precedents and/or Trace Error. If fixing all the topmost leftmost errors does not fix the rest of your errors on your worksheet, you may need to do it the hard way, from the bottom right upwards then leftwards; that is the slow but sure way to fix all errors.
  • Also, errors in your chart data will most likely plot as zeroes. This may be acceptable or desirable even (as in the current case). However, if too many lines (or curves) are returning to 0, it may indicate a logical flaw in the data -- or too many tiny values and then perhaps rescaling the chart is needed by inspecting the horizontal and vertical axes and changing them to zero in on the problem. Hover over or click on a datamarker on the series plot and then do a search in the proper column by value for that value, and identify its precedents.
    Error Returns Normal

Sources and Citations

  • "CRC Standard Curves and Surfaces", David von Seggern, 1993, CRC Press Inc., Boca Raton, FL. ISBN 0-8493-0196-3, Library of Congress Card Number 92-33596, [pp. 264 7.1.4 Spherical Helix]
  • One possible candidate source file for this article is "Karen's Worker.02.xlsx". See Chrys folder.

Article Info

Categories: Graphics | Microsoft Excel Imagery