https://en.wikipedia.org/wiki/Torus
A ring torus with aspect ratio 3, the ratio between the diameters of the larger (magenta) circle and the smaller (red) circle. In geometry, a torus ( pl.: tori or toruses) is a surface of revolution generated by revolving a circle in three-dimensional space one full revolution about an axis that is coplanar with the circle.

https://www.mathsisfun.com/geometry/torus.html
Note: Area and volume formulas only work when the torus has a hole! Like a Cylinder. Volume: the volume is the same as if we "unfolded" a torus into a cylinder (of length 2πR): As we unfold it, what gets lost from the outer part of the torus is perfectly balanced by what gets gained in the inner part.

https://mathworld.wolfram.com/Torus.html
An (ordinary) torus is a surface having genus one, and therefore possessing a single "hole" (left figure). The single-holed "ring" torus is known in older literature as an "anchor ring." It can be constructed from a rectangle by gluing both pairs of opposite edges together with no twists (right figure; Gardner 1971, pp. 15-17; Gray 1997, pp. 323-324). The usual torus embedded in three

https://www.merriam-webster.com/dictionary/torus
torus: [noun] a large molding of convex profile commonly occurring as the lowest molding in the base of a column.

https://www.omnicalculator.com/math/torus-volume
In addition to the these radii, the torus can also be expressed in the form of two radii such as inner (a) and outer radii (b) of the torus. Mathematically, that is: a = R - r b = R + r. The volume V of the said torus is: V = 2 * π² * r² * R The volume can also be written in terms of the inner and outer radii: V = 0.25 * π² * (b - a)²

https://mathmonks.com/torus
The formula is: Volume of a Torus. Volume (V) = 2πR × πr 2 = V = 2π 2 Rr 2. If a torus is cut and unfolded, it will take the shape of a cylinder, as shown in the diagram below. Torus Shaped Unfolded to Cylinder. The volume of a cylinder is πr 2 × length; here the length equals 2πR, where R = major radius of the torus.

https://www.vedantu.com/maths/torus
Surface Area of Torus Formula = 4 x π² x R x r. Similarly, volume of Torus is calculated as. V = 2 π² Rr². Volume of Torus Formula = V = 2 π² Rr². Facts to Remember. The value of in volume and surface area of the torus is constant and approximately equals 3.14 or 22/7. Two or more than two torus is known as tori.

https://math.stackexchange.com/questions/1352792/how-to-derive-the-3d-equation-of-a-torus
This leads to the parametric form of the torus: x(u, v) y(u, v) z(u, v) =(c + a cos(v)) cos(u), =(c + a cos(v)) sin(u), = a sin(v). What's left is verifying that this form is equivalent to the formula above. This is pretty straightforward. Hopefully, the parametric form provides a more natural perspective. Share.

https://encyclopediaofmath.org/wiki/Torus
A torus is a special case of a surface of revolution and of a canal surface. From the topological point of view, a torus is the product of two circles, and therefore a torus is a two-dimensional closed manifold of genus one. If this product is metrizable, then it can be realized in $ E ^{4} $ or in the elliptic space $ El ^{3} $ as a Clifford

https://math.stackexchange.com/questions/358825/parametrisation-of-the-surface-a-torus
A circle of radius a a centered at (b, 0) ( b, 0) in the plane xz x z has the parametric equation. x = a cos(θ) + b, z = a sin(θ), x = a cos. ⁡. ( θ) + b, z = a sin. ⁡. ( θ), with θ θ in the range [0, 2π] [ 0, 2 π] for a full circle. Now you rotate the plane xz x z around z z by x ← x cos(ϕ), y ← x sin(ϕ) x ← x cos. ⁡.

https://app.tor.us/
Torus Wallet is the easiest blockchain digital wallet available on the web. Take advantage of social logins like Google, Facebook, Twitter, Discord, Reddit and Email to instantly and securely manage your private keys, cryptocurrencies, digital tokens and DeFi transactions.

https://www.mathsisfun.com/definitions/torus.html
It usually looks like a ring. Illustrated definition of Torus: A 3d shape made by revolving a small circle (radius r) along a line made by a bigger circle (radius R).

https://en.wikipedia.org/wiki/Clifford_torus
Clifford torus. A stereographic projection of a Clifford torus performing a simple rotation. Topologically a rectangle is the fundamental polygon of a torus, with opposite edges sewn together. In geometric topology, the Clifford torus is the simplest and most symmetric flat embedding of the Cartesian product of two circles S1. a and S1.

https://archive.lib.msu.edu/crcmath/math/math/t/t200.htm
A torus is a surface having Genus 1, and therefore possessing a single `` Hole .''. The usual torus in 3-D space is shaped like a donut, but the concept of the torus is extremely useful in higher dimensional space as well. One of the more common uses of -D tori is in Dynamical Systems. A fundamental result states that the Phase Space

https://math.stackexchange.com/questions/103621/why-is-s1-times-s1-a-torus
16. Call a "torus" that geometric shape that "looks like" a doughnut. Frequently, one encounters the assertion that S1 ×S1 S 1 × S 1 is a "torus" where S1 S 1 is the unit circle. Now, if I think about this, I can understand the justification for calling this a torus, but I'm trying to understand how one would go about actually proving this.

https://simple.wikipedia.org/wiki/Torus
A ring torus. A circle spinning around a line to make a torus. A torus (plural: tori or toruses) is a tube shape that looks like a doughnut or an inner tube. In geometry, a torus is made by rotating a circle in three dimensional space. To make a torus, the circle is rotated around a line (called the axis of rotation) that is in the same plane

https://en.wikipedia.org/wiki/Algebraic_torus
Algebraic torus. In mathematics, an algebraic torus, where a one dimensional torus is typically denoted by , , or , is a type of commutative affine algebraic group commonly found in projective algebraic geometry and toric geometry. Higher dimensional algebraic tori can be modelled as a product of algebraic groups .

https://www.uniguide.com/torus
A torus is a three-dimensional geometric shape that resembles a donut, a life preserver ring, or a tire inner tube. The name is derived from Latin, in which it means a circular pillow, a strong rope, or a muscle. 1. When illustrated, the torus appears like a revolving circle in three-dimensional space that's revolving around an axis that is

https://math.stackexchange.com/questions/1926136/1-dimensional-and-n-dimensional-torus
A 2-dimensional torus, is formed by gluing $\left [ 0,1 \right ] \times \left [ 0,1 \right ]$. A 1-dimensional torus certainly requires only $\left [ 0,1 \right ]$. Is it a circle? Any help is appreciated. Thanks in advance. general-topology; Share. Cite. Follow

https://www.encyclopedia.com/science-and-technology/astronomy-and-space-exploration/astronomy-general/torus
Torus. A torus, in geometry, is a doughnut-shaped, three-dimensional figure formed when a circle is rotated through 360 ° about a line in its plane, but not passing through the circle itself. The word torus is derived from a Latin word meaning bulge. The plural of torus is tori. Another common example of a torus is the inner tube of a tire.

https://encyclopediaofmath.org/wiki/Algebraic_torus
[1] A. Borel, "Linear algebraic groups" , Benjamin (1969) MR0251042 Zbl 0206.49801 Zbl 0186.33201 [2] T. Ono, "Arithmetic of algebraic tori" Ann. of Math.(2), 74 : 1 (1961) pp. 101 139 MR0124326 Zbl 0119.27801 [3] T. Ono, "On the Tamagawa number of algebraic tori" Ann. of Math.(2), 78 : 1 (1963) pp. 47 73 MR0156851 Zbl 0122.39101

https://math.stackexchange.com/questions/1315349/surface-area-of-a-torus
7. Use Pappus theorem. If the radius of the transversal section of the torus is r r then its perimeter is 2πr 2 π r and Pappus theorem states that the surface of the torus (it is a revolution surface) equals A = 2πr ⋅ 2πR A = 2 π r ⋅ 2 π R where R R is the radius of rotation that generates the torus. In your case this is.

https://www.forbes.com/sites/startswithabang/2021/07/21/why-the-universe-probably-isnt-shaped-like-a-donut/
A torus, whose shape most commonly resembles a donut: the kind with a hole in the center. A visualization of a 3-torus model of space, where our observable Universe could be just a small