Robert P. Munafo, 2008 Mar 13.

Colloquial name for the portion of R2F(1/2B1) that is farthest away from the center of the Mandelbrot set; the definition varies but usually it refers to the area west of R2F(1/2B1)S. In extreme cases the term is used to refer only to R2t.

The smallest island of any given period is found in the utter west. Here is a table of the first few:

This sequence includes the John Dewey Jones midget. Each successive island is located about 4 times closer to the tip at -2.0+0i. Each island is about 1/16 the size, and 1/256 the area of the one before it. Here is the island with period 27:

leftmost period-27 island

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The book Mm - Much Ado About Nothing - Vol. 1, (A.G. Davis Philip, Adam Robucci, Michael Frame & Kenelm Philip, LC catalog number 91-092943) discusses these islands. The last midget in their sequence is R2F(1/2B1){FS[2]}x297S, with period 300, and its image is one of the deepest ever seen. To find the centers of these midgets, they discovered a general formula:

x = -2 + (6 (2j+1)² π²) / 4ⁿ⁺¹

where x is the real coordinate, n is the island's period, and j is a selector which should be zero.¹ The formula gives a location a bit to the left of the actual island. Higher values of j locate other islands.

This formula is not completely accurate for locating leftmost island of period N. As N increases, the location given by the formula tends to drift gradually to the left of the desired point. The following is much more accurate, and should properly center the view for N up to at least a few hundred:

center = -2 + 3π²/(2×4ⁿ) + 36.524n/16ⁿ - 100/16ⁿ
size = 200/16ⁿ

The first two terms are identical to the Robucci formula; the next term (with coefficient 36.524) cancels the gradual "drift". The last term is proportional to the size and therefore changing its coefficient will shift all islands the same amount in relation to the view.

See also easternmost point, northernmost point.

1 : http://classes.yale.edu/fractals/MandelSet/MandelScalings/HRScaling/LocatingMidgets.html "Hurwitz-Robucci scaling - Locating Midgets", from website of a Yale course titled Fractal Geometry.