Chip described a way to create a fractal. I paraphrase: Start with an equalateral triangle. Find the current point by picking a vertex at random. Put a mark at the current point (the vertex you just picked). Change the current point by picking a vertex at random and making the current point the midway point between the old point and the current point. Put a mark there and repeat. Do that about 50,000 times and see what you get. Or paste this code into a workbook and run SheetTriangle. Make sure you run it from Excel and not the VBE – it’s cooler to watch than just see the end result.

Dim CurrX As Double

Dim CurrY As Double

Dim Vertices(1 To 3, 1 To 2) As Double

Dim NextVert As Long

Dim i As Long

Dim wsh As Worksheet

Vertices(1, 1) = 128

Vertices(1, 2) = 1

Vertices(2, 1) = 1

Vertices(2, 2) = 227

Vertices(3, 1) = 256

Vertices(3, 2) = 227

Set wsh = ThisWorkbook.Worksheets.Add

wsh.Cells.RowHeight = 1.5

wsh.Cells.ColumnWidth = 0.17

‘Start at the third vertex

NextVert = 3

CurrX = Vertices(NextVert, 1)

CurrY = Vertices(NextVert, 2)

‘loop ten thousand times

For i = 1 To 50000

NextVert = Int(3 * Rnd + 1) ‘pick a random vertext

GetNewPoint CurrX, CurrY, Vertices(NextVert, 1), _

Vertices(NextVert, 2) ‘find the midway point

PlacePointWsh CLng(CurrX), CLng(CurrY), wsh ‘color a cell at that point

Next i

End Sub

Sub GetNewPoint(ByRef CurrX As Double, ByRef CurrY As Double, _

ByVal RandX As Double, ByVal RandY As Double)

CurrX = CurrX + ((RandX – CurrX) / 2)

CurrY = CurrY + ((RandY – CurrY) / 2)

End Sub

Sub PlacePointWsh(ByVal NewX As Long, ByVal NewY As Long, ByRef wsh As Worksheet)

wsh.Cells(NewY, NewX).Interior.Color = vbBlack

End Sub

I started doing this with shapes instead of colored cells. It didn’t look quite as good and I’d end up with 10,000 shapes on a sheet, which can’t be too stable. I changed to coloring cells and upped the iterations to 50,000, which is roughly 256 x 227. I don’t know if 50k is too many or too few, but it definitely gives you a good idea. The 227 is what I calculated the height in cells to be to make the legs of the triangle 256. You don’t need an equalateral triangle, though, it’s just the way it was first presented to me. I also heard that the randomness isn’t necessary, but I couldn’t make it work otherwise. I’m sure it’s possible, I’m just not up to it.

I asked Chip to whom credit was due, and he said he came up with it himself. I’m not too shocked, he’s a pretty smart guy. I think 1,000 monkeys on a 1,000 typewriters would have a better chance coming up with this than me. Some guy named Sierpinski beat him to it by about 90 years, albeit via a different method.

Fractals are beautiful aren’t they?

I really like animated fractals too, like zooming into a mandelbrot.

The method you describe is pretty much how my fractal landscape generator works.

After I created the formula page, I noticed how big the workbook got – about 2MB. Being wary of large downloads, I decided to create the formula page on workbook open.

A fractal generator generator.

Rob

Oh thanks a lot. That’s a half hour I’ll never get back.

Talking of mandelbrot…

http://www.andypope.info/fun/mandelbrot.htm

Although black and white is what sierpinski saw, it is more colourfull to whatch it with:

Sub PlacePointWsh(ByVal NewX As Long, ByVal NewY As Long, ByRef wsh As Worksheet)

wsh.Cells(NewY, NewX).Interior.Color = Choose(Int(7 * Rnd + 1), _

vbBlack, vbRed, vbBlue, vbGreen, vbYellow, vbMagenta, vbWhite)

End Sub