Search results
Results from the WOW.Com Content Network
Then there is a square centered at P which is entirely contained in the carpet. This square contains a smaller square whose coordinates are multiples of 1 / 3 k for some k. But, if this square has not been previously removed, it must have been holed in iteration k + 1, so it cannot be contained in the carpet – a contradiction.
Each face of the construction becomes a Sierpinski carpet, and the intersection of the sponge with any diagonal of the cube or any midline of the faces is a Cantor set. The cross-section of the sponge through its centroid and perpendicular to a space diagonal is a regular hexagon punctured with hexagrams arranged in six-fold symmetry. [8]
Shoelace scheme for determining the area of a polygon with point coordinates (,),..., (,). The shoelace formula, also known as Gauss's area formula and the surveyor's formula, [1] is a mathematical algorithm to determine the area of a simple polygon whose vertices are described by their Cartesian coordinates in the plane. [2]
Following Grünbaum and Shephard (section 1.3), a tiling is said to be regular if the symmetry group of the tiling acts transitively on the flags of the tiling, where a flag is a triple consisting of a mutually incident vertex, edge and tile of the tiling. This means that, for every pair of flags, there is a symmetry operation mapping the first ...
Algebra and Tiling: Homomorphisms in the Service of Geometry is a mathematics textbook on the use of group theory to answer questions about tessellations and higher dimensional honeycombs, partitions of the Euclidean plane or higher-dimensional spaces into congruent tiles.
A tiling with rectangles is a tiling which uses rectangles as its parts. The domino tilings are tilings with rectangles of 1 × 2 side ratio. The tilings with straight polyominoes of shapes such as 1 × 3, 1 × 4 and tilings with polyominoes of shapes such as 2 × 3 fall also into this category.
Truchet tiles are square tiles decorated with patterns so they do not have rotational symmetry; in 1704, Sébastien Truchet used a square tile split into two triangles of contrasting colours. These can tile the plane either periodically or randomly. [46] [47] An einstein tile is a single shape that forces aperiodic tiling. The first such tile ...
Comparison of 1 square yard with some Imperial and metric units of area The square yard ( Northern India : gaj , Pakistan : gaz ) is an imperial unit and U.S. customary unit of area . It is in widespread use in most of the English -speaking world, particularly the United States, United Kingdom, Canada, Pakistan and India.