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Tupper's self-referential formula is a formula defined by Jeff Tupper that, when graphed in two dimensions at a very specific location in the plane, can be "programmed" to visually reproduce the formula itself. It is used in various math and computer science courses as an exercise in graphing formulae. Although it is colloquially known as a "self-referential formula", this is actually a misnomer, because the image does not encode the constant K which is external data, and Tupper himself did not describe his formula that way.〔(【引用サイトリンク】url=https://shreevatsa.wordpress.com/2011/04/12/how-does-tuppers-self-referential-formula-work/ )〕 The formula was first published in his 2001 SIGGRAPH paper that discusses methods related to the (GrafEq ) formula-graphing program he developed. The formula is an inequality defined by: : where denotes the floor function, and is the modulo operation. Let ''k'' equal the following 543-digit integer: If one graphs the set of points (''x'', ''y'') in 0 ≤ ''x'' < 106 and ''k'' ≤ y < ''k'' + 17 satisfying the inequality given above, the resulting graph looks like this (note that the axes in this plot have been reversed, otherwise the picture comes out upside-down): The formula itself is a general-purpose method of decoding a bitmap stored in the constant ''k'', so it could actually be used to draw any other image. When applied to the unbounded positive range 0 ≤ y, the formula tiles a vertical swath of the plane with a pattern that contains all possible 17-pixel-tall bitmaps. One horizontal slice of that infinite bitmap depicts the drawing formula itself, but this is not remarkable, since other slices depict all other possible formulae that might fit in a 17-pixel-tall bitmap. Tupper has disseminated, via email, extended versions of his original formula that rule out all but one slice ((), (), ()). The constant ''k'' is a simple monochrome bitmap image of the formula treated as a binary number and multiplied by 17. If ''k'' is divided by 17, the least significant bit encodes the upper-right corner (''k'', 0); the 17 least significant bits encode the rightmost column of pixels; the next 17 least significant bits encode the 2nd-rightmost column, and so on, forming the image of the formula; so ''k'' holds a bitmap reproduced by the formula and is significant to have a subtle re-draw facility on appropriately scaled hardware. == See also == * Recursion * Quine (computing) * Strange loop * Bitmap 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Tupper's self-referential formula」の詳細全文を読む スポンサード リンク
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