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For instance, while all the cross-sections of a ball are disks, [2] the cross-sections of a cube depend on how the cutting plane is related to the cube. If the cutting plane is perpendicular to a line joining the centers of two opposite faces of the cube, the cross-section will be a square, however, if the cutting plane is perpendicular to a ...
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 ...
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.
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 ...
A principal -bundle, where denotes any topological group, is a fiber bundle: together with a continuous right action such that preserves the fibers of (i.e. if then for all ) and acts freely and transitively (meaning each fiber is a G-torsor) on them in such a way that for each and , the map sending to is a homeomorphism.
In mathematics, the Borromean rings [a] are three simple closed curves in three-dimensional space that are topologically linked and cannot be separated from each other, but that break apart into two unknotted and unlinked loops when any one of the three is cut or removed.
Similarly, the natural monomorphism Z/2Z → Z/4Z doesn't split even though there is a non-trivial morphism Z/4Z → Z/2Z. The categorical concept of a section is important in homological algebra , and is also closely related to the notion of a section of a fiber bundle in topology : in the latter case, a section of a fiber bundle is a section ...
A two-dimensional Poincaré section of the forced Duffing equation. In mathematics, particularly in dynamical systems, a first recurrence map or Poincaré map, named after Henri Poincaré, is the intersection of a periodic orbit in the state space of a continuous dynamical system with a certain lower-dimensional subspace, called the Poincaré section, transversal to the flow of the system.