How to Acquire the Black Mosaic Tile Image via Excel

The Tutorial

1. Open a new Microsoft Excel workbook and name three worksheets in it: "Data", "Chart" and "Saves". Depending on which version of Excel you're using, when you get to the steps concerning creating the new chart, you'll either use Chart Wizard and a Chart page will be created for you, or you will cut and paste the chart from the Data sheet. Furthermore, the steps below have been copied to a large extent from Make a Square of Spherical Helixes, so if you'd like to save some time and have completed that article, you can do a SAVE AS on it to a new workbook, title it appropriately, and save it to a logical file folder. Look for the words MODIFIED or NEW to indicate changes from the original text.
2. MODIFIED and NEW: On the Data sheet, enter the following Defined Variable Names in Row 1, (without quotation marks): A1: "AdjRows", B1: "GMLL", C1: "Factor", D1: "N", E1: "Multer1", F1: "Multer2", G1: "p", H1: "NUM", I1: "SPHEROIDS2", J1: "Factor2", K1: "SHRINK2", M1: "SHRINK3"
3. MODIFIED: Select range A1:M2 and Align Center. Make the cell borders bold and individual in the color of your choice. Do Insert Name Create in Top Row. In p, in cell G1, Insert New Comment, "0,P to P,P then P,P to P,0, then P,0 to 0,0, then 0,0 to 0,P." That forms a square when we Fill in series.
4. MODIFIED: In Row 2 enter the following Defined Variable values and formulas (w/o quotes): A2: 2880, B2: "=-(1-SQRT(5))/2", C2: "=IF(ISNA(VLOOKUP(SPHEROIDS, LOOKER,2)),0.125,VLOOKUP(SPHEROIDS,LOOKER,3))", D2: 1,E2: .5, F2:.5, G2:1, H2: 1, I2:16, J2: .125, K2:.95. M2: .65.
5. NEW: In Row 3, enter: A3: BASE, B3: 1712, C3: SPHEROIDS, D3: 31.25, E3: TIP, F3: "=25680*2*6*PI()/15", G3: SHRINK, H3: 1. Command Select A3:B3 and Insert Name Define name BASE to cell $B$3. Command Select C3:D3 and Insert Name Define name SPHEROIDS to cell $D$3. Command Select E3:F3 and Insert Name Define name TIP to cell $F$3. Command Select G3:H3 and Insert Name Define name SHRINK to cell $H$3.
6. MODIFIED/NEW: Across Row 4, into columns A:N please type the following Column Headers (without quotes): A4: "Base t", B4: "c", C4: "cos", D4: "sin", E4: "SQR CHART x", F4: "SQR CHART y", G4: "x of p", H4: "y of p", I4: "COS2", J4: "SIN2", K4: "OUTRING X", L4: "OUTRING Y", M4: "INRING X", N4: "INRING Y". Select cell range A4:M4 and Align Center and Underline text.
7. MODIFIED: In cell B5, enter w/o quotes "=(BASE*24/SPHEROIDS)". In column B, select cell range B6:B2877. With cell B6 as the active highlighted cell, input the formula, without quotes, "=B5" and then do Edit Fill Down. The constant 1314.82 should appear now in the range from B5 to B2877, as the formula copied the value each time from the cell just above itself.
8. MODIFIED: Select cell A5 and Insert Name Define "Top" for cell $A$5. Input the following formula without quotes into cell A5: "=IF(ODD(SPHEROIDS)=SPHEROIDS,0,TIP)". The value 64,540.88 should appear. Do Format Cell Number Custom without the brackets ["top" 0.00] and widen the column if need be.
9. MODIFIED: Select cell range A6:A2887 and with cell A6 as the highlighted active cell, input the formula, "=(A5+(-top*2)/(AdjRows))" and do Edit Fill Down. The value -968,113.19 should appear in cell A1444 and the value -969,458.73 should appear in cell A1445. By multiplying the top times 2, we took care of both the negative as well as the positive values while decrementing  each time with the negative sign.
10. MODIFIED:Select cell range C5:C2887 and with C5 as the highlighted active cell, enter the formula, "=SPHEROIDS/36*COS((ROW()-5)*PI()/180*Factor)". Do Edit Fill Down. We multiply by pi/180 to convert to degrees from pi radians. The value in (ROW()-5) relative to C5 is 0; therefore, we are taking the cosine of 0.
11. MODIFIED: Select cell range D5:D2887 and with D5 as the highlighted active cell, enter the formula, "=SPHEROIDS/36*SIN((ROW()-5)*PI()/180*Factor)". Do Edit Fill Down. The value 1.0000 should appear in cell C1445 and the value 0.0000 should appear in cell D1445 after selecting range C5:H1445 and Cell Format Number for 4 decimal places.
12. MODIFIED: Select cell range E5:E2887 and with E5 the active cell, enter the formula w/o quotes "=((SIN(A5/(B5*2))*GMLL*SHRINK*COS(A5)*GMLL*SHRINK*(COS(A5/(B5*2)))*SHRINK*GMLL)+G5*SHRINK)^(N-0)". Do Edit Fill Down. That is a zero at the very end, not an OH. We will not be using Multer1 or Multer2 -- they are used for warping and other effects inside or outside this formula. Basically, this is the formula for a spherical helix in a ring defined by adding C5 instead of G5, but by adding G5 we're creating a square instead. "GMLL" stands for "Golden Mean Long Leg" and it has a tendency to keep things proportionate no matter how much they're scaled up or down in size, so each term is multiplied by this factor. Notice that for the x values, the basic scheme is SIN*COS*COS. We are raising to the power of N, currently set to 1, which you may play with and note the interesting effects of the N=.5 or N=2 or N=3 on the space inside of 1^2 in the graph space. It would seem that what they taught us about 1 raised to any power equaling 1 is not true, when in fact it is, but it sure doesn't seem that way here.
13. MODIFIED: Select cell range F5:F2887 and with F5 the hi-lighted active cell, enter the formula w/o quotes "=((SIN(A5/(B5*2))*SHRINK*GMLL*SIN(A5)*SHRINK*GMLL*(COS(A5/(B5*2)))*SHRINK*GMLL)+H5*SHRINK)^(N-0)", then do Edit Fill Down. This would form a ring of spheres if we added D5 instead of H5 near the end of the formula because columns C and D, being the cosine and sine, form a circle. That option might necessitate a Factor of .125 and of course new Edit Fill Downs for the formulas in columns E and F. Notice that for the y values, the basic scheme is SIN*SIN*COS. The final COS element is the Z or 3rd dimension for both E and F. The graph is a fractal and is neither truly 2-dimensional nor 3-dimensional per the actual staining medium on the paper. You may think of it like an architectural drawing where the height dimension has been taken down to nearly zero, but not quite zero.
14. MODIFIED: We'll now be following the little note in cell G1: "0,P to P,P then P,P to P,0, then P,0 to 0,0, then 0,0 to 0,P." There are 4 sides of the square to do in about 2324 cells, so that's 581 cells per side (I should have stopped at 576 if you want to correct things). We're going to do one side at a time: first the x's, then the y's.
15. NEW: Enter to cell O1 w/o quotes "SpheroidSurroundRows". Enter to O2 w/o quotes "=2880/SPHEROIDS" and do Insert Name Define Name SpheroidSurroundRows to cell $O$2.
16. MODIFIED: Enter 0 into cell G5. Select cell range G6:G585 and with cell G6 as the highlighted active cell, enter the formula "=p/(SpheroidSurroundRows*SPHEROIDS/5)+G5". Do Edit Fill Down. Select cell range H5:H585 and with cell H5 the highlighted active cell, enter the formula "=p" and Edit Fill Down. This completes the first part of the graph or the first side, from "0,P to P,P". Select cell range G5:H365 and Format Cell Border Black Bold Outline.
17. MODIFIED Select cell range G586:G1161 and with cell G586 as the hi-lighted active cell, enter the formula, "=p". Do Edit Fill Down. Select cell range H586:H1161 and with cell H586 as the highlighted active cell, enter the formula, "=-p/(SpheroidSurroundRows*SPHEROIDS/5)+H585" -- the minus sign is very important. Do Edit Fill Down. Select cell range G586:H1161 and Format Cell Border Red Bold Outline. This completes the second side, from "P,P to P,0,".
18. MODIFIED: Select cell range G1162:G1746 and with cell G1162 as the highlighted active cell, enter the formula "=-p/(SpheroidSurroundRows*SPHEROIDS/5)+G1161" and do Edit Fill Down. Select cell range H1162:H1746 and with cell H1162 as the active highlighted cell, enter the formula, "=0". Do Edit Fill Down. This completes the third side of the square, from "P,0 to 0,0". Select cell range G1162:H1746 and Format Cell Fill Canary Yellow and Border Bold Dark Blue.
19. NEW: Select cell range G1747:G2330 and with cell G1747 active, enter "=0" and Edit Fill Down. Select cell range H1747:H2330 and with cell H1747 highlighted and active, enter the formula "=p/(SpheroidSurroundRows*SPHEROIDS/5)+H1746" and do Edit Fill Down. The last fourth leg or side of the square is done, "0,0 to 0,P."
20. NEW: COS2 -- Edit Go To cell range I5:i2887 and with I5 the active cell, enter the formula w/o quotes, "=SPHEROIDS2/36*COS((ROW()-5)*PI()/180*Factor2)" and Edit Fill Down.
21. NEW: SIN2 -- Edit Go To cell range J5:J2887 and with J5 the active cell, enter the formula w/o quotes, "=SPHEROIDS2/36*SIN((ROW()-5)*PI()/180*Factor2)" and Edit Fill Down.
22. NEW: OUTRING X -- Edit Go To cell range K5:K2887 and with K5 the active cell, enter the formula w/o quotes, "=0.5+((SIN(A5/(B5*2))*GMLL*SHRINK2*COS(A5)*GMLL*SHRINK2*(COS(A5/(B5*2)))*SHRINK2*GMLL)+I5*SHRINK2)^(N-0)" and Edit Fill Down.
23. NEW: OUTRING Y -- Edit Go To cell range L5:L2887 and with L5 the active cell, enter the formula w/o quotes, "=0.5+((SIN(A5/(B5*2))*SHRINK2*GMLL*SIN(A5)*SHRINK2*GMLL*(COS(A5/(B5*2)))*SHRINK2*GMLL)+J5*SHRINK2)^(N-0)" and Edit Fill Down.
24. NEW: INRING X -- Edit Go To cell range M5:M2887 and with M5 the active cell, enter the formula w/o quotes, "=0.5+((SIN(A5/(B5*2))*GMLL*SHRINK3*COS(A5)*GMLL*SHRINK3*(COS(A5/(B5*2)))*SHRINK3*GMLL)+I5*SHRINK3)^(N-0)" and Edit Fill Down.
25. NEW: INRING Y -- Edit Go To cell range N5:N2887 and with N5 the active cell, enter the formula w/o quotes, "=0.5+((SIN(A5/(B5*2))*SHRINK3*GMLL*SIN(A5)*SHRINK3*GMLL*(COS(A5/(B5*2)))*SHRINK3*GMLL)+J5*SHRINK3)^(N-0)" and Edit Fill Down. And those are all zeroes at the far right hand end of each formula, not OH's.
26. MODIFIED: Select cell Q4 and type LOOKER.

    1. Edit Go To cell range Q5:Q68 and enter 1 into cell Q5 and do Edit Fill Series Columns Linear Step Value 1, OK.
    2. Edit Go To cell range R5:R68 and enter .125 and Edit Fill Down.
    3. Edit Go To cell range S5:S68 and enter "=Q6*$S$34/$Q$34" into cell S5 and Edit Fill Down.
    4. Select cell Q34 and enter .125 and make it font red fill yellow, as an input cell..
    5. Edit Go To cell range Q5:S68 and Insert Name Define Name Looker to cell range $Q$5:$S$68 and format fill lightest blue and border dark blue bold outline.
27. NEW: It's always a good idea to Insert New Comment and input the original formulas and values of the cells as you'll probably tend to experiment and change them. It's much easier having the original formula or value right there as pasted from the copy from the formula bar than trying to find it in this article. I urge you to do that now while you check for errors against this articles inputs, as it shows no errors. If you do have errors, see also the WARNINGS section below about how to fix them if they are not obvious. And a column full of zeroes should be considered an error as well, if it's the entire column. Please also bear in mind that some variables have been set up to play with later -- it's just that we're at the basic level at this point. Some variable could probably be better named, eg. Power for N, or ShrinkorExpand for Shrink.

Explanatory Charts, Diagrams, Photos

1. (dependent upon the tutorial data above)

1. NEW: Create the Chart.

   1. Check x of p and y of p: Edit Go To cell range G5:H2330 and do Charts from the Ribbon or Chart Wizard, All/Other, Scattered - Smoothed Line Scatter. The result should be a direct square from {0,1} thru each side of 1 unit length up to point {1,1} and back to {0,1} (approx.) {0,1} actually occurs in cells G2322:H2322, but the discrepancy doesn't matter and has to do with what the spheroids need to come out looking right anyway, not a straight line. If that chart checks out OK, delete it, otherwise use it to correct the formulas in the x of p and y of p columns.
   2. Edit Go To E5:F2330 and do Charts from the Ribbon or Chart Wizard, All/Other, Scattered - Smoothed Line Scatter. Copy or Cut the resulting chart and paste it to the upper left hand corner of the Chart worksheet. Do Chart Layout and get rid of Axes, Grid lines and Legend. Grab the lower right hand corner and expand the chart into a nice large square shape at about J45. You must wait until the cursor changes to a double-headed arrow before you can grab and resize the chart.
   3. Do menu item Chart Add Data and in response, click on the Data worksheet and select cell range K5 to L2877. Mine never works right. I get two series, neither of which is right, There's a green one that has spheroids, so I click on that and edit its formula in the formula bar until it reads, "=SERIES(,DATA!$K$5:$K$2887,DATA!$L$5:$L$2877,2)" (make sure both say 2887). Then click on the other series and delete it. Click in the plot area.
   4. Do menu item Chart Add Data again and in response, click on the Data worksheet and select cell range M5 to N2877. Mine never works right. I get two series, neither of which is right, There's a green one that has spheroids, so I click on that and edit its formula in the formula bar until it reads, "=SERIES(,DATA!$M$5:$M$2877,DATA!$N$5:$N$2877,3)" (make sure both say 2887). Then click on the other series and delete it.
   5. Click in the chart Plot Area and do Chart Layout (at left) Plot Area, Format Selection. Select Fill, Black, 3-D Format Top Hard Edge 16 pt Height 16 pt for Bevel and under Depth & Surface Surface Metal.
   6. Click in the Chart Area surrounding the Plot Area and do Chart Layout Chart Area Format Selection. 3-D Format: Top should be Hard Edge 16 pt Width 16 pit Height. Depth and Surface Surface should be Dark Edge. Fill should be black. Shadow should be checked and Outer 45 degrees black 100% size, 4 pt blur, 111 pt distance and 57% transparency (or so).
   7. Do Chart Layout Series 1 Format Selection. Color is Black, smoothed line, 2.5 pt, 0% transparency; Shadow is checked 21 degrees Outer Color 2nd lightest Grey, 100% size, 4 pt blur, 20 pt distance, 57% transparency; Glow and Soft Edges should be 3rd grey down, 2 pt size, 0& transparency, 1 pt soft edges. Change order: Move down to 3 or bottom.
   8. Do Chart Layout Series 2 Format Selection. Color is color wheel, darkened, about 5.6 from center slightly to left (7-10%) from direct blue and color bar at right is down at about 57-60%, so this is a dark blue, kind of muddy, tending a bit more towards green than violet and not all the way out at the edge RGB is 22, 46, 109. Line weight is 3 point, smoothed line., no shadow, glow is yellow 1 pt transparency 1% soft edges 0 pt. Order is topmost, # (furthest backgrounded).
   9. Do Chart Layout Series 3 Format Selection.. RGB is 24, 18, 32 color wheel very dark violet purple (near black) -- on the violet chart bar it's about 90% towards the bottom and halfway out in the very dark dark purple circle. Smoothed Line, Line Weight 2.5 pts. Shadow is checked Outer 315 degrees yellow 100% size, 4 pt blur, 3 pt distance and 0% transparency. Glow and Soft Edges is White, 2 pt size, 25% transparent, 0 pt soft edges.

2. Original mosaic from the slideshow:

3. Version done as I wrote the article (the one at top of the article was done before I wrote the article -- this is a 3rd version):

Helpful Guidance

1. Make use of helper articles when proceeding through this tutorial:

   1. See the article How to Create a Spirallic Spin Particle Path or Necklace Form or Spherical Border for a list of articles related to Excel, Geometric and/or Trigonometric Art, Charting/Diagramming and Algebraic Formulation.
   2. 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.