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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 ...
In the mathematical field of topology, a section (or cross section) [1] of a fiber bundle is a continuous right inverse of the projection function. In other words, if E {\displaystyle E} is a fiber bundle over a base space , B {\displaystyle B} :
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.
Cross section of a Pythagorean cup being filled: at B, it is possible to drink all the liquid in the cup; but at C, the siphon effect causes the cup to drain. A Pythagorean cup looks like a normal drinking cup, except that the bowl has a central column in it, giving it a shape like a bundt pan. The central column of the bowl is positioned ...
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 ...
Hexagonal cross-section of a level-4 Menger sponge. (Part of a series of cuts perpendicular to the space diagonal.) The n {\displaystyle n} th stage of the Menger sponge, M n {\displaystyle M_{n}} , is made up of 20 n {\displaystyle 20^{n}} smaller cubes, each with a side length of (1/3) n .
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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.