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If a plane intersects a solid (a 3-dimensional object), then the region common to the plane and the solid is called a cross-section of the solid. [1] A plane containing a cross-section of the solid may be referred to as a cutting plane. The shape of the cross-section of a solid may depend upon the orientation of the cutting plane to the solid ...
1. A cone and a cylinder have radius r and height h. 2. The volume ratio is maintained when the height is scaled to h' = r √ π. 3. Decompose it into thin slices. 4. Using Cavalieri's principle, reshape each slice into a square of the same area. 5. The pyramid is replicated twice. 6. Combining them into a cube shows that the volume ratio is 1:3.
In algebraic geometry, a lemniscate (/ l ɛ m ˈ n ɪ s k ɪ t / or / ˈ l ɛ m n ɪ s ˌ k eɪ t,-k ɪ t /) [1] is any of several figure-eight or ∞-shaped curves. [ 2 ] [ 3 ] The word comes from the Latin lēmniscātus , meaning "decorated with ribbons", [ 4 ] from the Greek λημνίσκος ( lēmnískos ), meaning "ribbon", [ 3 ] [ 5 ...
Sections are studied in homotopy theory and algebraic topology, where one of the main goals is to account for the existence or non-existence of global sections. An obstruction denies the existence of global sections since the space is too "twisted". More precisely, obstructions "obstruct" the possibility of extending a local section to a global ...
A necessary and sufficient condition for (, /,,) to form a fiber bundle is that the mapping admits local cross-sections (Steenrod 1951, §7). The most general conditions under which the quotient map will admit local cross-sections are not known, although if G {\displaystyle G} is a Lie group and H {\displaystyle H} a closed subgroup (and thus a ...
Cross-stitch counted-thread embroidery. Knitted mathematical objects include the Platonic solids, Klein bottles and Boy's surface. The Lorenz manifold and the hyperbolic plane have been crafted using crochet. [2] [3] Knitted and crocheted tori have also been constructed depicting toroidal embeddings of the complete graph K 7 and of the Heawood ...
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p −1 (1 − ε q p −1). Thus the elements of the subspace { ε q : q ∈ H} do not have inverses. This subspace is called an ideal in ring theory. It happens to be the unique maximal ideal of the ring of dual numbers. The group of units of the dual number ring then consists of numbers not in the ideal.