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An a × b rectangle can be packed with 1 × n strips if and only if n divides a or n divides b. [ 15 ] [ 16 ] de Bruijn's theorem : A box can be packed with a harmonic brick a × a b × a b c if the box has dimensions a p × a b q × a b c r for some natural numbers p , q , r (i.e., the box is a multiple of the brick.) [ 15 ]
The most efficient way to pack different-sized circles together is not obvious. In geometry, circle packing is the study of the arrangement of circles (of equal or varying sizes) on a given surface such that no overlapping occurs and so that no circle can be enlarged without creating an overlap. The associated packing density, η, of an ...
Happily the context of 51 and 52, together with the base, mid-line, and smaller triangle area (which are given as 4 + 1/2, 2 + 1/4 and 7 + 1/2 + 1/4 + 1/8, respectively) make it possible to interpret the problem and its solution as has been done here. The given paraphrase therefore represents a consistent best guess as to the problem's intent ...
Mathematics of paper folding. Map folding for a 2×2 grid of squares: there are eight different ways to fold such a map along its creases. The discipline of origami or paper folding has received a considerable amount of mathematical study. Fields of interest include a given paper model's flat-foldability (whether the model can be flattened ...
However, the blue triangle has a ratio of 5:2 (=2.5), while the red triangle has the ratio 8:3 (≈2.667), so the apparent combined hypotenuse in each figure is actually bent. With the bent hypotenuse, the first figure actually occupies a combined 32 units, while the second figure occupies 33, including the "missing" square.
A wallpaper group (or plane symmetry group or plane crystallographic group) is a mathematical classification of a two-dimensional repetitive pattern, based on the symmetries in the pattern. Such patterns occur frequently in architecture and decorative art, especially in textiles, tiles, and wallpaper. The simplest wallpaper group, Group p 1 ...
Pattern Blocks. Pattern Blocks are a set of mathematical manipulatives developed in the 1960s. The six shapes are both a play resource and a tool for learning in mathematics, which serve to develop spatial reasoning skills that are fundamental to the learning of mathematics. Among other things, they allow children to see how shapes can be ...
Given four indices i 1 ≤ i 2 and j 1 ≤ j 2, the set E(i 1,i 2,j 1,j 2) contains the indices of all rectangles whose bottom-left corner is in the rectangle [x i 1,x i 2] X [y p (j1),y p (j2)]. A cutting pattern is a guillotine pattern if and only if, for all quadruplets of indices i 1 ≤ i 2 and j 1 ≤ j 2, at least one of the following ...
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