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The dot pattern is a two-dimensional barcode; the most common is the proprietary Anoto dot pattern. In the Anoto dot pattern, the paper is divided into a grid with a spacing of about 0.3 mm, a dot is printed near each intersection offset slightly in one of four directions, a camera in the pen typically records a 6 x 6 groups of dots.
Thus a single line can be drawn connecting all nine dots—which would appear as three lines in parallel on the paper, when flattened out. [18] It is also possible to fold the paper flat, or to cut the paper into pieces and rearrange it, in such a way that the nine dots lie on a single line in the plane (see fold-and-cut theorem). [17]
In January 2002, she folded a 4,000-foot-long (1,200 m) piece of toilet paper twelve times in the same direction, debunking a long-standing myth that paper cannot be folded in half more than eight times. [21] [22] The fold-and-cut problem asks what shapes can be obtained by folding a piece of paper flat, and making a single straight complete ...
Hexagonal paper shows regular hexagons instead of squares. These can be used to map geometric tiled or tesselated designs among other uses. Isometric graph paper or 3D graph paper is a triangular graph paper which uses a series of three guidelines forming a 60° grid of small triangles. The triangles are arranged in groups of six to make hexagons.
A radius-5 circle overlaid with a grid of dots in the pattern of a dot planimeter. When counting dots near the boundary of the shape as 1/2, there are 69 interior dots and 20 boundary dots for an estimated area of 79, close to the actual area of 25 π ≈ 78.54.
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Dots and boxes is a pencil-and-paper game for two players (sometimes more). It was first published in the 19th century by French mathematician Édouard Lucas, who called it la pipopipette. [1] It has gone by many other names, [2] including dots and dashes, game of dots, [3] dot to dot grid, [4] boxes, [5] and pigs in a pen. [6]
Initially, paper was ruled by hand, sometimes using templates. [1] Scribes could rule their paper using a "hard point," a sharp implement which left embossed lines on the paper without any ink or color, [2] or could use "metal point," an implement which left colored marks on the paper, much like a graphite pencil, though various other metals were used.
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