What is on the screen
The Sierpinski triangle is what you get by removing the middle quarter of a triangle, then the middle quarter of each remaining triangle, forever. Recursion mode draws exactly that, 3depth triangles. Chaos game mode arrives at the same shape by a route that looks like it should not work: a point hops halfway towards a randomly chosen corner, over and over, and the trail of hops fills in the fractal — never the holes, because no hop can land in one. Randomness here is not noise, it is the mechanism for visiting every part of the shape evenly.
Change the number of corners and the shrink factor r and the same game produces the Sierpinski carpet, the Vicsek cross, and pentagonal and hexagonal relatives. Or take a square with r = ½ and no restriction: it fills in solidly, dimension 2. Add the rule that the same corner cannot be chosen twice in a row, and a fractal reappears out of the same square, dimension log 3 / log 2 again.
What “1.585 dimensions” means, and how the tool measures it
Lay a grid over a shape and count the cells it touches. Halve the cell size and count again. A line touches twice as many cells; a filled square touches four times as many. A shape of dimension D touches 2D times as many, so D is the slope of log(count) against log(1/cell size). The Sierpinski triangle is three half-size copies of itself, so it touches 3× as many: D = log 3 / log 2 = 1.5850. The carpet is eight third-size copies: log 8 / log 3 = 1.8928. The plot above shows the counts and the fitted line, and the table shows the counts next to the ideal 3k, 8k, or 5k sequence for the shape.
The 0.017 that was not the triangle’s fault
The experiment this tool is built from measured the equilateral triangle with 4 million chaos-game points, grids from 4×4 to 256×256, and got 1.568 — 0.017 short of theory. Ten random seeds moved the answer by only ±0.0003, twenty times more points left it at 1.5684, and grids 16 times finer left it at 1.5675. None of the obvious suspects. The cause was the grid: an equilateral triangle’s slanted sides cut across square cells, so every small triangle bleeds into two or three neighbours, and the counts came out as 12, 36, 106, 310, 900, 2763, 8304 instead of 9, 27, 81, 243, 729, 2187, 6561.
Switch the shape to right-isosceles, aligned to the grid and the same chaos game with the same counting code returns the ideal sequence exactly, 3k at every level, and the fitted dimension reads 1.5850. Switch back, and it drifts. In the experiment, moving the equilateral triangle to a different position in the square even flipped the sign of the error, to 1.614. The measurement had been reporting the relationship between the triangle and the grid, not the triangle.
It is not that alignment fixes everything, either. Keep the aligned triangle but change the grid refinement from halving to thirds and the counts stop following any pattern (6, 42, 273, 1581, …) and the fit lands around 1.66 to 1.67 — the experiment, with finer grids and ten times the points, got 1.6345. Either way it is 0.05 or more off. The triangle shrinks by halves, so a grid that shrinks by halves meshes with it; a grid that shrinks by thirds does not. For the carpet it is the other way round: thirds match, halving does not. In a measurement of an unknown shape you never know which refinement meshes — which is why the honest use of this tool is the range of values you get as you change the grid, not any single one.
Measuring your own image
Choose Measure my image and drop in a picture. It is converted to black-and-white with the threshold slider (dark pixels count as “ink”), fitted into the square, and box-counted on the same grids. Coastlines come out around 1.2 to 1.3, a straight-edged logo close to 1, a photograph of texture close to 2. Two cautions. First, the result depends on the range of grid sizes — below the line width everything looks two-dimensional, above the image size everything looks like a dot — so read the plot and see whether the points actually lie on a line. Second, before trusting the number on an unknown image, feed the tool a shape whose answer you know: the aligned triangle should read 1.585 and a filled rectangle 2.000. If it does not, the problem is the measurement setup, not the image.