recursal/Fanatic-Fandom huggingface dataset:
Dodecagon

A **dodecagon** is a 2-dimensional polygon with twelve edges. The Bowers acronym for a dodecagon is **dog**.

## Structure and Sections

The regular dodecagon has each angle at 150 degrees. It can be seen as a truncated hexagon.

### Hypervolumes

* vertex count =

12

{\displaystyle 12}
* edge length =

12
l

{\displaystyle 12 l}
* surface area =

3

(

2

3

)

l
2

{\displaystyle 3\left(2 + \sqrt{3}\right) l^2}

### Subfacets

* 12 points (0D)
* 12 line segments (1D)
* 1 dodecagon (2D)

### Radii

* Vertex radius:

2

6

2

l

{\displaystyle \frac{\sqrt{2}+\sqrt{6}}{2}l}
* Edge radius:

2

3

2

l

{\displaystyle \frac{2+\sqrt{3}}{2}l}

### Vertex coordinates

The vertices of a regular dodecagon centered at the origin, of side 2, can be given by all sign changes of:

* (1+√3, 1+√3)
* (1, 2+√3)
* (2+√3, 1)

### Related shapes

* Dual: Self-dual
* Vertex figure (and second diagonal): Line segment, length

2

6

2

{\displaystyle \frac{\sqrt{2}+\sqrt{6}}{2}}
* Third diagonal: [[Length

1

3

{\displaystyle 1+\sqrt{3}}
* Fourth diagonal: Length

3

2

6

2

{\displaystyle \frac{3\sqrt{2}+\sqrt{6}}{2}}
* Fifth diagonal: Length

2

3

{\displaystyle 2+\sqrt{3}}
* Longest diagonal: Length

2

6

{\displaystyle \sqrt{2}+\sqrt{6}}

## See Also

t
0

{
12
}

{\displaystyle t\_0 \{12\}}

t
1

{
12
}

{\displaystyle t\_1 \{12\}}

t

0
,
1

{
12
}

{\displaystyle t\_{0,1} \{12\}}

t
0

{
6
}

{\displaystyle t\_0 \{6\}}

t
1

{
6
}

{\displaystyle t\_1 \{6\}}

t

0
,
1

{
6
}

{\displaystyle t\_{0,1} \{6\}}
Cartesian product

The **Cartesian product** of two sets A and B is the set of all possible ordered pairs (a, b) such that a is an element of A and b is an element of B (using set-builder notation, Cartesian products can be written as

A
×
# B
{
(
a
,
b
)

a
∈
A

and

b
∈
B
}

{\displaystyle A \times B = \{(a, b)|a \in A \ \textrm{and} \ b \in B\}}

). The cardinality of the Cartesian product of two sets is always the product of the cardinalities of the two sets.

The Cartesian product of two shapes A and B, is a shape whose coordinates are (a, b), where a is a coordinate of shape A and b is a coordinate of shape B. The dimensionality of the resulting shape is the sum of the dimensionalities of A and B.

Cartesian products of multiple sets are known as "n-fold Cartesian products" or "n-ary Cartesian products", and will result in a set of n-tuples. *Cartesian powers* are a unary operation defined by repeated cartesian products. The Cartesian power of a set is written

A
n

{\displaystyle A^n}

, where A is a set and n is some non-negative integer.

A
n

{\displaystyle A^n}

 means

A
×
A
.
.
.
×
A

⏟

n

times

{\displaystyle \underbrace {A \times A ... \times A}\_{n \ \textrm{times}}}

, and there is a bijection between the cardinality of

A
n

{\displaystyle A^n}

 and the cardinality of the set of functions from a set with n elements and A. If A has a cardinality of 2, then the cardinality of

A
n

{\displaystyle A^n}

 is equal to the cardinality of the power set of a set with n elements.

Repeated cartesian products of n Euclidean lines will result in n-dimensional Euclidean spaces.

The Cartesian product of a shape (typically a polytope) and a line segment is the shape's prism.

Repeated cartesian products of a line segment will create a hypercube.

Repeated cartesian products of hyperballs will create a rotatope. If hyperspheres are to be included in the Cartesian products, then toratopes will be the result.

Taking the Cartesian product of two shapes (typically polytopes) results in a duoprism.
The Cartesian product of some shape and a point is the original shape.

The Cartesian product of any set (which can be a shape) and the empty set is the empty set. This is because the cardinality of the empty set is 0, and any number (in this case, the cardinality of the set/shape) multiplied by 0 is 0, including the cardinality of the continuum.
Conductance

**Conductance** is a physical quantity measuring the ease of passing an electric current through a component. It has units of [current]2[mass]-1[length]-2[time]3.

The idealised circuit element possessing only conductance is called the resistor. Conductance is related to the voltage and current across a resistor by Ohm's law,

# G

I
V

{\displaystyle G = \frac{I}{V}}

. This means it is the reciprocal of resistance.

## Conductances

 3.3×10-2 S
 ∞ S
Class S star

**Class S stars** are a rare variant of red giant stars characterized by roughly equal amounts of oxygen and carbon in their atmospheres and the presence of strong zirconium oxide bands rather than titanium oxide bands (a defining characteristic of Class M stars) in their spectra. Class S stars have a temperature range from 1,800 to 4,000 K—comparable in temperature to stars of class L ("deep red"), M ("red"), and late K ("orange").

There are two distinct classes of Class S stars—**intrinsic** and **extrinsic**. Intrinsic Class S stars are asymptotic giant branch strars that undergo thermal pulsing from helium flashes which will grow stronger as the star evolves. Intrinstic S-type stars are technetium stars and this feature is used to distinguish between them and extrinsic S-type stars. Intrinsic S-type stars are generally slightly cooler than their extrinsic counterparts—Mira variable S-type stars average around 2,300 K and non-Mira S-type stars average around 3,100 K. The coolest Class S stars are around 1,800 K in temperature. Extrinstic Class S stars have a temperature range from 3,150 to 4,000 K and are typically red giant branch stars rather than asymptotic giant branch stars. Extrinsic S-type stars are generally less luminous and longer-lived than their intrinsic counterparts.

## References

* Various; S-type star - Wikipedia, the free encyclopedia

## See Also

**Brown dwarf classes**: **Y** · **T** · **L** · **M**
**Wolf-Rayet and carbon star classes**: ****S**** · **C** · **W**
**Stellar remnant classes**: **D** · **N** · **Ω**
Brane

A **brane** is an object similar to a quantum string, but with any number of dimensions rather than the single dimension of a string. X-dimensional branes trace out (X+1)-dimensional worldvolumes.

Collisions between branes can create new protoverses, which explode outwards into universes.

According to the models of Brane Cosmology our universe, along with possibly many others, would exist trapped inside branes with the same number of dimensions of such universes. Our universe would exist in a 3 spacial dimensions brane and so on.
4-Dimensional space

A **4-Dimensional Space** is a space in which each point requires a quadruplet of numbers to describe its position. Examples of 4-dimensional spaces include a space with 3 spatial dimensions and 1 dimension used to find extents in our universe, spaces required to represent quaternions or coquaternions, and 4-manifolds with special curvatures.

It is the first hyperrealm, meaning a space with 4 or more dimensions.

## Types of 4-Dimensional spaces

Examples of 4-dimensional spaces are listed below.

### Tetraspherical

A tetrasphere is a four dimensional surface with positive curvature, and the four-dimensional equivalent of a 2-dimensional sphere. It bounds a 5-dimensional space called a pentorb.

### Euclidean

A Euclidean flune is a four-dimensional space with zero curvature. It is a flune that follows the postulates of Euclidean geometry.

### Hyperbolic

A hyperbolic flune is a four-dimensional flune with negative curvature.

### Glomitubic

An infinite glomitube is a surface created from the Cartesian product of a three-dimensional glome and a Euclidean line.

## Verses

A verse with four dimensions is called a fluneverse. Spatiotemporally, our universe is thought to be a fluneverse, though models of string theory made for phenomenology require more dimensions in our universe (multiverse, if you consider our universe to be the space localized on a D3 brane) to be consistent.
Colour drive

A **colour drive**, also known as a **gluon assist drive**, is a spacecraft propulsion method that uses the strong nuclear force to alter the motion of the spacecraft. Ordinarily, this is impossible, because the strong nuclear force only applies over very small ranges due to the colour confinement property of hadrons; however, in universes with compatible laws of physics, or with sufficient femtoengineering potentially available to very advanced civilizations, this may be feasible.

One advantage of the strong interaction is that it is constant over distance, meaning it can provide a large amount of acceleration regardless of the distance scales involved.

Operation of the drive is detectable by the spray of gluons and highly radioactive hadrons that it would emit due to the interaction energies involved in the strong nuclear force.
Cardverses

A Cardverse is a type of Verse, whose Soupcount index corresponds to a cardinal number. Each cardverse contains a larger Hierarchy of Verses, than the one below. This is different from the other Soupcount Archverses, for example an ω+2 Verse would hold ω+1 verses, ω verses and verses with a finite index, א‎0+3 is still א‎0. So while an ω+3 Verse is on a higher Level, than an ω verse, the hierarchies they hold are still both of the same length. An ω1 Verse would hold a larger hierarchy of verses, than any Verse below it, thus making it a cardverse.

## The different types of Cardverses

### Low level Cardverses

On the finite Level, there is no real difference between ordinal numbers and cardinal numbers. That means, that every Verse below the Omniverse can be considered a Cardverse. However, there is a difference between those cardverses and the ones beyond the Omniverse. Normally Cardverses are always Lodeverses as well, as they cannot be reached by just going up a finite amount of steps in the archverse chain. Finite level Cardverses however, can be reached that way. Therefore, while these verses are technically classified as Cardverses, a distinction must still be made between the lower level Cardverses and the ones beyond the Omniverse,

### Higher Level Cardverses

The Omniverse marks the end of the low level Cardverses and the beginning of the high level- also called the true Cardverses. The Soupcount ordinal index of the Omniverse is ω, corresponding to the cardinal א‎0. Following the Omniverse is the so called Great Omniverse, with a Soupcount ordinal index of ω1 corresponding to the cardinal א‎1, followed by the ω2 verse, corresponding to א‎2 and so on. Eventually all Alephs are used up. The Verse containing all of these verses would correspond to the inaccessible cardinal. This pattern goes on forever: Whenever a cardinal is used up, the next cardinal will be used and whenever a set of cardinals gets used up, like with the alephs, a new, larger set of cardinals is used to number the Archverses. This Hierarchy never stops, as there will always be a larger Cardinal to use to number the next verse.

## Beyond the Cardverses

The Verse beyond all Cardverses is the Barrelplex, as it is not simply the next Verse in the chain, but the Verse containing all hierarchies that can be made using this method. To go beyond the Barrelplex, you cannot simply "go higher", as all simple ways of going higher are already contained within the Barrelplex. The only way to truly transcend the Barrelplex is to have a verse, thats bigger than itself, a so called Selfverse.
Bilinski rhombic dodecahedron

A **Bilinski rhombic dodecahedron** is a variation on a standard rhombic dodecahedron. The arrangement of vertices, edges and faces is identical to the standard rhombic dodecadron, though the shapes of the rhombi are different; the diagonal lengths on a Bilinski rhombic dodecahedron are a and φa, with φ being the golden ratio (1 + √5)/2.

It can also be made from removing the middle band from the rhombic triacontahedron, and then you get the rhombic icosahedron, and you remove the middle band from that, and you now have a Bilinski rhombic dodecahedron.
Cross polytope

A **cross-polytope** or **orthoplex** is a shape constructed by repeatedly taking the bipyramid of a point. Each hypercube has four simplices of the lower dimension surrounding each peak (e.g. an octahedron has four triangles around each vertex, a hexadecachoron has three tetrahedra around each vertex).
Agricultural growth regime

The **agricultural growth regime** is the growth regime that begins when a civilization begins altering the environment around them through introducing more land-efficient energy sources than the existing ones, and then managing this source. This process is known as agriculture. For example, a biological terrestrial agriculturalist may begin replacing wild flora with fields of crops.

The increase in the carrying capacity of the land, and the possibility for increased population to continue raising the carrying capacity, massively accelerates the growth rate. Economic size is still tied to population and carrying capacity, though this carrying capacity can now be increased. Doubling times are typically on the order of a thousand years, and this regime typically lasts tens of thousands of years.
Dodekasphere

The **dodekasphere** is the **12-dimensional** surface made from getting the set of all points equidistant from another in 13-dimensional space. It makes up the surface of a tridekorb.

## Structure and Sections

The cross-section of a dodekasphere looks like a point expanding to a hendekasphere and then back to a point.

### Hypervolumes

* surdokon dodekavolume =

π

6

332640

l

12

{\displaystyle \frac{{\pi}^{6}}{332640} l^{12}}
4-position

**4-position** is a physical quantity in special relativity with units of position. Like all 4-vectors, it has one timelike component and three spacelike components. These are written as

X

=

(

c
t
,

r
→

)

{\displaystyle \mathsf{X} = \left( ct, \vec{r} \right)}

, where the timelike component is the time (or duration) of the event and the spacelike component is the position (or length) of the event.

The square of the four-position is the same in all inertial reference frames. This quantity is called the spacetime interval.
Chariklo

Universe → Pisces–Cetus Supercluster Complex → Laniakea Supercluster → Virgo Supercluster → Local Group → Milky Way Galaxy → Orion-Cygnus Arm → Gould Belt → Local Bubble → Local Interstellar Cloud → Solar System → **Chariklo**

## Physical Structure

### Diameter

254.7 km

**Chariklo** is the largest known centaur in our Solar System and the first known minor planet in our Solar System to have a ring system. Chariklo orbits the Sun between Saturn and Uranus, is an inner-grazer of Uranus, and, like most centaurs, has a dynamically unstable orbit. Chariklo’s orbit is more stable than Chiron’s. Chariklo’s ring system consists of two rings and it is thought that Chariklo might have currently undiscovered moons shepherding the rings.
Cubinder

A **cubinder** is a four-dimensional rotatope. It is the duoprism formed from the product of a square and a disc, and can also be considered to be a prism with a cylinder as the base. It is composed of four cylindrical cells joined by a curved 3-D cell.

## Structure and Sections

The cubinder has two views for sections. It can look either as an unchanging cylinder or a square expanding to a cube before shrinking back.

### Subfacets

* 4 circles (1D)
* 4 discs (2D)
* 4 tubes (2D)
* 4 cylinders (3D)
* 1 square torus (3D)
* 1 cubinder (4D)
Cubic honeycomb

A yellow honeycomb set against a black background.

A **cubic honeycomb** is a 3-dimensional uniform honeycomb covering the Euclidean realm made of cubic cells, joined four to each edge. It is also the only regular honeycomb in three dimensions.

Its Bowers acronym is **chon**.

## Structure and Sections

The cubic honeycomb is its own runcination.

## See Also

Template:Variant Nav 4 3 4

{
2
,
3
,
4
}

{\displaystyle \{2,3,4\}}

{
3
,
3
,
4
}

{\displaystyle \{3, 3, 4\}}

{
4
,
3
,
4
}

{\displaystyle \{4,3,4\}}

{
5
,
3
,
4
}

{\displaystyle \{5,3,4\}}

{
6
,
3
,
4
}

{\displaystyle \{6,3,4\}}

{
4
,
2
,
4
}

{\displaystyle \{4,2,4\}}

{
4
,
3
,
4
}

{\displaystyle \{4,3,4\}}

{
4
,
4
,
4
}

{\displaystyle \{4,4,4\}}

{
4
,
5
,
4
}

{\displaystyle \{4,5,4\}}

{
4
,
6
,
4
}

{\displaystyle \{4,6,4\}}

{
4
,
3
,
2
}

{\displaystyle \{4,3,2\}}

{
4
,
3
,
3
}

{\displaystyle \{4, 3, 3\}}

{
4
,
3
,
4
}

{\displaystyle \{4,3,4\}}

{
4
,
3
,
5
}

{\displaystyle \{4,3,5\}}

{
4
,
3
,
6
}

{\displaystyle \{4,3,6\}}

## References

1. ↑ https://bendwavy.org/klitzing/incmats/chon.htm
Complex projective line

A **Riemann sphere** or **complex projective line** is a complex 1-dimensional manifold that consists of the complex line plus a point at ∞. Points on its surface represent extended complex numbers, the points near ∞ are numbers with larger absolute values, the points near ∞'s antipode 0 are numbers with smaller absolute values, and points on the equator are numbers with an absolute value of 1, numbers that satisfy

e

θ
i

{\displaystyle e^{\theta i}}

 where

θ

{\displaystyle \theta}

 is a real number.

The extended complex line projected on the sphere is very useful to complex analysis because they allow division by zero to be well-behaved.
Altarca

An **Altarca**,plural **Altarcae** (from "*alt*ernate" and the Latin word "*arca*" meaning "box"), is a self-contained cosmology that exists within the Box but does not necessarily have any connection to any other cosmological hierarchy or the elements therein. Any written or otherwise defined cosmology forms an Altarca around itself, no matter how large or small.

The Barrelplex is the "default" Altarca assumed to be in play by this wiki's shared cosmological elements; it is marked by a simple recursion at the lower levels, counting upwards, via powersetting or similar operations, from Universes through Omniverses and the like. Other Altarcae do not need to have any of these traits. Some can be closed loops, where the "largest" -verse is contained by the "smallest", while others branch off into multiple loops that are then surpassed by something else. Certain Altarcae are "small" and have only a finite number of levels within; others are "large" and complex enough to surpass even the usual notions of post-infinites or the concept of meaningful arrangement itself.

Altarcae can contain themselves, larger Altarca cosmologies, ultimata such as the Box or the Omniumverse, or anything else that forms a cosmology, but they are ultimately still contained within the Box. In this way, they can be compared to the Imaginata of the Imaginarium, except mapped to cosmologies rather than imagined realms.

An entity or structure that is somehow not part of any cosmology can be said to exist within Metempiric Space, but doing so without inadvertently defining oneself as a cosmology of sorts (and thus an Altarca) is difficult.

This wiki's Cosmology Tier page effectively details the contents of various Altarcae.
Big Crunch

A **Big Crunch** is a possible endstate of an initially expanding universe where the scale factor reaches to zero in finite time after the universe stops expanding and begins contracting.

One way this can occur is a universe filled with a field that has an energy density that dilutes faster than dark energy (i.e. energy density increases with scale factor). Fields that do so include matter, radiation, and ultralight. For a universe dominated with a field with initial energy density

Ω

n

{\displaystyle \Omega\_{n}}

 that varies with scale factor as

a

−
n

{\displaystyle a^{-n}}

 and minimal curvature (for a 3D realmverse: if

# n
1

{\displaystyle n=1}

, the universe is domain wall-dominated; if

# n
2

{\displaystyle n=2}

, the universe is cosmic string-dominated; if

# n
3

{\displaystyle n = 3}

, the universe is matter-dominated; if

# n
4

{\displaystyle n = 4}

, the universe is radiation-dominated; and if

4
<
n

{\displaystyle 4<n}

, the universe is ultralight-dominated), the Friedmann equation is

H
2

H
0
2

=

Ω

n

a

−
n

{\displaystyle \frac{H^2}{H\_0^2} = \Omega\_{n} a^{-n}}

 and therefore the scale factor

a

{\displaystyle a}

 as a function of time is

# a

4

−

1
n

(

n
2

Ω
n

(

H
0

t
−

a
0

)
2

)

1
n

{\displaystyle a = 4^{-\frac{1}{n}} ( n^2 \Omega\_n (H\_0 t - a\_0 )^2 )^\frac{1}{n} }

. This reaches zero at time

# t

a
0

H
0

{\displaystyle t = \frac{a\_0}{H\_0}}

.

To an observer inside the universe, the Big Crunch represents the end of everything - no force is able to keep structures apart, so galaxies, solar systems, atoms, and eventually spacetime itself are crushed together and the observer's own observable universe will be compressed into an infinitesimally small point reminiscent to how it was during the universe's beginning.

Following a Big Crunch, it is possible that the collapsed universe’s spacetime could re-expand and a new universe can potentially be born in a Big Bang event after the previous universe underwent its Big Crunch event. An event like this is known as a Big Bounce.

It is likely that our universe will not end in a Big Crunch event as its expansion has been observed to be accelerating. This suggests that a Big Rip or heat death event is far more likely to occur in the future.
Asteroid belt

Universe → Pisces–Cetus Supercluster Complex → Laniakea Supercluster → Virgo Supercluster → Local Group → Milky Way Galaxy → Orion-Cygnus Arm → Gould Belt → Local Bubble → Local Interstellar Cloud → Solar System → **Asteroid belt**

The **asteroid belt** is the disc of irregular small bodies that roughly orbits between Mars and Jupiter in the Solar System.

## Major Asteroids

Around half of the mass of the asteroid belt is concentrated in the four largest asteroids: Ceres, Vesta, Pallas and Hygiea.

## Asteroid Families

   The asteroid belt can be subdivided into multiple families that represent clusters of asteroids with similar orbits.

One division between families is formed by the Kirkwood gaps - asteroids that have an integer ratio of their semimajor axis with Jupiter are nudged out of that orbit by resonance, which creates divisions that are clearly visible on a plot of asteroid orbits.

Families within this subdivisions are formed by collisions - when two asteroids collide, they release a very large number of small asteroids that all share a similar orbit. Because of this, bodies in an asteroid family tend to have similar compositions.
Big Rip

A **Big Rip** is a possible endstate of an expanding universe where the scale factor tends to infinity in finite time.

One way this can occur is a universe filled with a field that has an energy density that dilutes slower than dark energy (i.e. energy density increases with scale factor). This field is called phantom energy. For a universe dominated phantom energy with initial energy density

Ω

−
n

{\displaystyle \Omega\_{-n}}

 that varies with scale factor as

a
n

{\displaystyle a^n}

 and minimal curvature, the Friedmann equation is

H
2

H
0
2

=

Ω

−
n

a

n

{\displaystyle \frac{H^2}{H\_0^2} = \Omega\_{-n} a^{n}}

 and therefore the scale factor as a function of time is

# a

4

1
n

(

1

n
2

Ω

−
n

(

H
0

t
−

a
0

)

2

)

1
n

{\displaystyle a = 4^\frac{1}{n} \left( \frac{1}{n^2 \Omega\_{-n} \left( H\_0 t - a\_0 \right)^2 } \right)^\frac{1}{n}}

; this reaches infinity at a time

# t

a
0

H
0

{\displaystyle t = \frac{a\_0}{H\_0}}

.

To an observer inside the universe, the Big Rip represents the end of everything - no force is able to keep structures bound, so galaxies, solar systems, atoms, and eventually spacetime itself are ripped apart. To an observer outside the universe, the Big Rip represents the universe becoming infinitely energetic; this provides a rich source of resources for civilisations with phantom energy mining technology.
Class Y star

A **Class Y star** is the dimmest type of brown dwarf. Many are classed as **sub-brown dwarfs** that may not be large enough to undergo deuterium fusion, at less than 13 Jupiter masses. A typical Class Y star has a temperature of around 300 - 400 K, a mass of around 6-25 times that of Jupiter, and a similar radius to Jupiter. Many less massive Class Y stars are often classified as large rogue planets.

## Properties of Y-type brown dwarfs

Similar to how methane absorption is used as the defining characteristic for Class T brown dwarfs, ammonia absorption is used as the defining characteristic for Class Y brown dwarfs. Unlike classes M and L and like class T, the Class Y category consists entirely of objects incapable of hydrogen fusion. The transition between Class T and Class Y brown dwarfs may not entirely have to do with the temperature of the objects and it is possible that other factors such as surface gravity, vertical mixing, clouds, and metallicity can affect an object's placement.

## Examples

* WISE 0855−0714, a Y4 dwarf and the coldest known brown dwarf with a temperature of about 250 K
* These are the 7 Y-type stars discovered by J. Davy Kirkpatrick et al. in 2012:
	+ WISE 0146+4234, a Y0 dwarf
	+ WISE 0350−5658, a Y1 dwarf
	+ WISE 0359−5401, a Y0 dwarf
	+ WISE 0535−7500, a Y1 dwarf
	+ WISE 0713−2917, a Y0 dwarf
	+ WISE 0734−7157, a Y0 dwarf
	+ WISE 2220−3628, a Y0 dwarf

## See also

**Brown dwarf classes**: ****Y**** · **T** · **L** · **M**
**Wolf-Rayet and carbon star classes**: **S** · **C** · **W**
**Stellar remnant classes**: **D** · **N** · **Ω**

## References

1. ↑ Various; Brown dwarf - Wikipedia, the free encyclopedia
2. ↑ Various; Stellar classification - Wikipedia, the free encylopedia
3. ↑ Dupuy, T. J., Kraus A. L.; Distances, Luminosities, and Temperatures of the Coldest Known Substellar Objects
4. ↑ Various; WISE 0855−0714 - Wikipedia, the free encyclopedia
5. ↑ Simbad database; WISEA J085510.74-071442.5
6. ↑ Kirkpatrick, J. D., Gelino, C. R., Cushing, M. C. et al.; Further Defining Spectral Type "Y" and Exploring the Low-mass End of the Field Brown Dwarf Mass Function
Boltzmann Brain

A **Boltzmann Brain** is a type of intelligent being, usually defined as a Cosmic Entity of sorts, formed from mathematical or physical chaos rather than the normal organic processes which generally give rise to life. Although extremely unlikely, random changes in the quantum structure of a universe may give rise to a self-aware entity; this process never occurs in almost all universes, but in the infinitely large higher-level -verses it occurs infinitely often. Therefore, there are an infinite number of Boltzmann Brains in existence.

The bare minimum requirement for a Brain is that an entity is capable of self awareness, and as such, a Brain could be significantly less intelligent than a human. However, it is also possible that they are in fact immensely intelligent, having formed with a fully-developed neural system far more advanced than those of humans, deities, or some cosmic entities.
Apeirogonal prism

An **apeirogonal prism** is a prism with an apeirogon as the base. It is a semiregular Euclidean planar tiling that be constructed by truncating an apeirogonal hosohedron.

Its Bowers acronym is **azip**.

## Structure and Sections

### Hypervolumes

* vertex count =

ℵ
0

{\displaystyle \aleph\_0}
* edge length =

∞
l

{\displaystyle \infty l}
* surface area =

∞

l
2

{\displaystyle \infty l^2}
* surcell volume =

∞

l
3

{\displaystyle \infty l^3}

### Subfacets

* ℵ0 points (0D)
* ℵ0 line segments (1D)
* ℵ0 squares (2D)
* 2 apeirogons (2D)
* 1 **apeirogonal prism** (3D)

## See Also

* Polytope Wiki. "Apeirogonal prism"

t

0
,
1

{
2
,
2
}

{\displaystyle {t}\_{0,1} \{2,2\}}

t

0
,
1

{
2
,
3
}

{\displaystyle {t}\_{0,1} \{2,3\}}

t

0
,
1

{
2
,
4
}

{\displaystyle {t}\_{0,1} \{2,4\}}

t

0
,
1

{
2
,
5
}

{\displaystyle {t}\_{0,1} \{2,5\}}

t

0
,
1

{
2
,
6
}

{\displaystyle {t}\_{0,1} \{2,6\}}

t

0
,
1

{
2
,
7
}

{\displaystyle {t}\_{0,1} \{2,7\}}

t

0
,
1

{
2
,
8
}

{\displaystyle {t}\_{0,1} \{2,8\}}

t

0
,
1

{
2
,
∞
}

{\displaystyle {t}\_{0,1} \{2,\infty\}}

t

0
,
1

{
2
,

π
i

λ

}

{\displaystyle {t}\_{0,1} \{2,\frac{\pi i}{\lambda}\}}

## References

1. ↑ https://bendwavy.org/klitzing/incmats/azip.htm
Bipyramid

A **bipyramid**, also known as a **tegum**, is a polytope constructed by joining together two pyramids of some given polytope by their bases. The elements of a tegum are two pyramids for each of the base's elements. The bipyramid itself does not contain the base of the pyramids joined together as an element.

## List of regular bipyramids

### 3 dimensions

* Triangular bipyramid
* Octahedron (Square bipyramid)
* Pentagonal bipyramid
* Hexagonal bipyramid
* Heptagonal bipyramid
* Octagonal bipyramid

### 4 dimensions

* Tetrahedral bipyramid
* Cubic bipyramid
* Hexadecachoron (Octahedral bipyramid)
* Dodecahedral bipyramid
* Icosahedral bipyramid
Class F star

A **Class F star** is the fourth-brightest type of main sequence star. They are typically colored white or yellow-white. They make up roughly 3% of main-sequence stars.

A typical Class F star has a mass of 1.2 (1.04 to 1.4) solar masses, a radius of 1.3 (1.15 to 1.4) solar radii, a luminosity of 3 (1.5 to 5) solar luminosities, a surface temperature of 6,750 K (6,000 K to 7,500 K), and a lifespan of 5 billion years.

While these stars have lifespans a bit shorter than the Sun, they are likely candidates for planets and life with habitable zones around 1.5 AU away, though perhaps they might become too bright and hot later in their lives for intelligent life to develop.

## See Also

**Brown dwarf classes**: **Y** · **T** · **L** · **M**
**Wolf-Rayet and carbon star classes**: **S** · **C** · **W**
**Stellar remnant classes**: **D** · **N** · **Ω**

## References

* Cram, Lawrence E., and Leonard V. Kuhi. *FGK Stars and T Tauri Stars*. NASA, 1989.
* Mamajek, E., *A Modern Mean Dwarf Stellar Color and Effective Temperature Sequence*. 2019
Cell

The **cells** of a polytope are the three-dimensional constituents of said polytope, three-dimensional subfacets. The cells of a polychoron compose its boundary. Polyhedra are composed of a single cell.

Cells are the facets of a polychoron, the ridges of a polyteron, and the peaks of a polypeton.

The total volume of a polytope's cells is called its surcell volume, or just volume if it is of a polyhedron.

## See Also

...Spire · Peak · Ridge · Facet · Polytope
Complex plane

The **complex plane** is a term that is commonly used to refer to a space with one real dimension and one imaginary dimension. However, that space, also known as an Argand plane, actually has only one complex dimension and should be called a complex line. Strictly speaking, a true complex plane is a space with two complex dimensions, and coordinates on said space need to be located with a pair of complex numbers.
Cantellated hecatonicosachoron

720 pentagons

600 octahedra

The **cantellated hecatonicosachoron**, the **cantellated 120-cell**, or, in Bowers' naming system, the **srahi**, is a uniform polychoron created by cantellating a hecatonicosachoron. It is composed of 120 rhombicosidodecahedral cells, 1200 triangular prism cells, and 600 octahedral cells, and its Schläfli symbol is

t

0
,
2

{

5
,
3
,
3

}

{\displaystyle t\_{0,2}\left\{5,3,3\right\}}

.

## Structure

Two rhombicosidedecahedra, two triangular prisms, and one octahedron meet at each wedge-shaped vertex. This shape is convex.

## See Also

* Polytope Wiki. "Small rhombated hecatonicosachoron".
* Bowers, Jonathan. "Category 6: Sphenoverts".

t
0

{
5
,
3
,
3
}

{\displaystyle t\_0 \{5,3,3\}}

t
1

{
5
,
3
,
3
}

{\displaystyle t\_1 \{5,3,3\}}

t
2

{
5
,
3
,
3
}

{\displaystyle t\_2 \{5,3,3\}}

t
3

{
5
,
3
,
3
}

{\displaystyle t\_3 \{5,3,3\}}

t

0
,
1

{
5
,
3
,
3
}

{\displaystyle t\_{0,1} \{5,3,3\}}

t

1
,
2

{
5
,
3
,
3
}

{\displaystyle t\_{1,2} \{5,3,3\}}

t

2
,
3

{
5
,
3
,
3
}

{\displaystyle t\_{2,3} \{5,3,3\}}

t

0
,
2

{
5
,
3
,
3
}

{\displaystyle t\_{0,2} \{5,3,3\}}

t

1
,
3

{
5
,
3
,
3
}

{\displaystyle t\_{1,3} \{5,3,3\}}

t

0
,
1
,
2

{
5
,
3
,
3
}

{\displaystyle t\_{0,1,2} \{5,3,3\}}

t

1
,
2
,
3

{
5
,
3
,
3
}

{\displaystyle t\_{1,2,3} \{5,3,3\}}

t

0
,
3

{
5
,
3
,
3
}

{\displaystyle t\_{0,3} \{5,3,3\}}

t

0
,
2
,
3

{
5
,
3
,
3
}

{\displaystyle t\_{0,2,3} \{5,3,3\}}

t

0
,
1
,
3

{
5
,
3
,
3
}

{\displaystyle t\_{0,1,3} \{5,3,3\}}

t

0
,
1
,
2
,
3

{
5
,
3
,
3
}

{\displaystyle t\_{0,1,2,3} \{5,3,3\}}
Antiprism

An **antiprism** is any shape with two identical copies of one shape on the top and bottom, and an alternating band of triangles in the middle. For example, an Icosahedron is an example of an antiprism because it has two corners where 5 triangles meet each on the top and bottom, and an alternating strip of triangles in the middle. A proper antiprism, though, would have a single polygon on the top and bottom.

## **Some well known examples**

* Icosahedron
* Polygon Antiprism Family (where the tops and bottoms are any regular polygon)
Circle pair

A **circle pair** is a one dimensional shape constructed from a pair of two disconnected circles. This shape is found as the boundary of some higher dimensional shapes, such as the two-dimensional tube.

## Knot Theory

In knot theory, embeddings of the circle pair into the Euclidean realm make up the 2-component links. These are distinct in that no link can be smoothly deformed into another without intersecting itself at some point

### Unlink

The simplest link, the unlink, is the link made up of a circle pair where the components are completely disconnected and seperate.

The unlink can be seperated out into two distinct unknots.

### Hopf Link

The Hopf link is the simplest nontrivial link, made up of a circle pair where the two circles are linked together exactly once but are otherwise undistorted.

The Hopf link can be seperated out into two distinct unknots.

### Solomon's Knot

Solomon's knot is a link made up of a circle pair in which the circles are doubly linked.

Solomon's knot can be seperated out into two distinct unknots

### Whitehead Link

The Whitehead link is a link made up of a circle pair, with one of the circles twisted into a figure-eight shape intertwined with a larger circle which goes through the two loops of the figure-eight.

The Whitehead link can be seperated out into two distinct unknots; the inner knot, twisted into a figure-eight shape, is only unable to be deformed into an unknot normally because it is linked with the outer knot which constrains it.
4-velocity

4-velocity is a quantity in special relativity related to classical velocity. It is a four-vector, defined as the proper time derivative of the 4-position (i.e. the rate of change of an object's position, as measured in that object's rest frame).

Since

X

=

(

c
t
,

x
→

)

{\displaystyle \mathsf{X} = \left( ct, \vec x \right)}

,

U

=

∂

∂
τ

(

c
t
,

x
→

)

=
γ

∂

∂
t

(

c
t
,

x
→

)

{\displaystyle \mathsf{U} = \frac{\partial}{\partial \tau} \left( ct, \vec x \right) = \gamma \frac{\partial}{\partial t} \left( ct, \vec x \right)}

 (by the definition of proper time), so

U

=
γ

(

c
,

v
→

)

{\displaystyle \mathsf{U} = \gamma \left( c, \vec v \right)}

.

The square of 4-velocity is

U

.

U

=

γ
2

(

v
2

−

c
2

)

=

v
2

−

c
2

1
−

v
2

c
2

=
−

c
2

{\displaystyle \mathsf{U}.\mathsf{U} = \gamma^2 \left(v^2 - c^2\right) = \frac{v^2 - c^2}{1 - \frac{v^2}{c^2}} = -c^2}

, which is the same in all reference frames.
Apeirogonal hosohedron

An **apeirogonal hosohedron** is a tiling composed of infinitely many digonal faces, all sharing the same two vertices. In normal Euclidean space, it is degenerate as the vertices are at infinity.

## See Also

{
2
,
0
}

{\displaystyle \{2,0\}}

{
2
,
1
}

{\displaystyle \{2,1\}}

{
2
,
2
}

{\displaystyle \{2,2\}}

{
2
,
3
}

{\displaystyle \{2,3\}}

{
2
,
4
}

{\displaystyle \{2,4\}}

{
2
,
5
}

{\displaystyle \{2,5\}}

{
2
,
6
}

{\displaystyle \{2,6\}}

{
2
,
7
}

{\displaystyle \{2,7\}}

{
2
,
8
}

{\displaystyle \{2,8\}}

{
2
,
∞
}

{\displaystyle \{2,\infty\}}

{
2
,

π
i

λ

}

{\displaystyle \{2,\frac{\pi i}{\lambda}\}}

Template:Variant Nav infty 2

{
2
,
∞
}

{\displaystyle \{2,\infty\}}

{
3
,
∞
}

{\displaystyle \{3,\infty\}}

{
4
,
∞
}

{\displaystyle \{4,\infty\}}

{
5
,
∞
}

{\displaystyle \{5,\infty\}}

{
6
,
∞
}

{\displaystyle \{6,\infty\}}

{
7
,
∞
}

{\displaystyle \{7,\infty\}}

{
8
,
∞
}

{\displaystyle \{8,\infty\}}

{
∞
,
∞
}

{\displaystyle \{\infty,\infty\}}

{

π
i

λ

,
∞
}

{\displaystyle \{\frac{\pi i}{\lambda},\infty\}}
AB-Matter

**AB-Matter** is a form of matter that can be produced using femtotechnology held together using the strong nuclear force, rather than the electromagnetic forces of typical matter.

This matter is then arranged in a mesh configuration, with the permeability of the AB-Matter to various external objects depending on the mesh size. A sufficiently small mesh can block gamma radiation; a sufficiently large one can pass through matter, since the thickness of a string of AB-Matter is on the order of a nucleon, much smaller than the spacing between molecules.

## References

1. ↑ Bolonkin A.A., Femtotechnology. AB-matter. Properties, Possibility Production and Applications
Dodecagram

A **dodecagram** is a concave star polygon created with twelve vertices and twelve edges, with every five points connected. It is the only star polygon with twelve sides.

## See Also

t
0

{

6
5

}

{\displaystyle t\_0 \{\frac{6}{5} \}}

t
1

{

6
5

}

{\displaystyle t\_1 \{\frac{6}{5} \}}

t

0
,
1

{

6
5

}

{\displaystyle t\_{0,1} \{\frac{6}{5} \}}
Algebraic line

The **algebraic line** is the shape created from taking all of the points on the complex line that correspond to the **algebraic numbers**, numbers that are the roots of non-constant polynomials with rational coeffiecients. It is a proper subset of the complex line, and the rational line is a subset of the algebraic line. The shape that corresponds to the set of all complex numbers that aren't in the algebraic line is known as the transcendental line.
Bowers acronym

A **Bowers acronym** is a short name assosciated with a polytope coined by Jonathan Bowers. Bowers coined the names as the real names for many polytopes are very long. Bowers acronyms are derived from pieces of the polytope's long name (for example, "**tes**" comes from tesseract which is a 4 dimensional structure) and some polytopes have the same long and short name (for example, square and cube).

The following lists of polytopes also list the polytopes' corresponding Bowers acronym.

* List of Uniform Polygons
* List of Uniform Polyhedra
* List of Uniform Compound Polyhedra
* List of Johnson Solids
* List of Uniform Polychora
* List of Scaliform Polychora
* List of Uniform Polytera
Decagon

A **decagon** is a 2-dimensional polygon with ten edges. The Bowers acronym for a decagon is **dec**.

## Structure and Sections

The regular decagon has each angle at 144 degrees. It can be seen as a truncated pentagon.

### Hypervolumes

* vertex count =

10

{\displaystyle 10}
* edge length =

10
l

{\displaystyle 10l}
* surface area =

5
2

5
2

5

l
2

{\displaystyle \frac{5}{2}\sqrt{5+2\sqrt{5}}l^2}

### Subfacets

* 10 points (0D)
* 10 line segments (1D)
* 1 decagon (2D)

### Radii

* Vertex radius:

1

5

2

l

{\displaystyle \frac{1+\sqrt{5}}{2}l}
* Edge radius:

5
2

5

2

l

{\displaystyle \frac{\sqrt{5+2\sqrt{5}}}{2}l}

### Vertex coordinates

The vertices of a regular decagon of side 2 can be given by all changes of sign of:

* (1, √(5+2√5))
* ((3+√5)/2, √((5+√5)/2))
* (1+√5, 0)

### Related shapes

* Dual: Self-dual
* Conjugate: Decagram
* Vertex figure: Line segment, length

5

5

2

{\displaystyle \sqrt{\frac{5++\sqrt{5}}{2}}}
* Third diagonal: Length

3

5

2

{\displaystyle \frac{3+\sqrt{5}}{2}}
* Fourth diagonal: Length

5
2

5

{\displaystyle \sqrt{5+2\sqrt{5}}}
* Long diagoal: Length

1

5

{\displaystyle 1+\sqrt{5}}

## See Also

t
0

{
10
}

{\displaystyle t\_0 \{10\}}

t
1

{
10
}

{\displaystyle t\_1 \{10\}}

t

0
,
1

{
10
}

{\displaystyle t\_{0,1} \{10\}}

t
0

{
5
}

{\displaystyle t\_0 \{5\}}

t
1

{
5
}

{\displaystyle t\_1 \{5\}}

t

0
,
1

{
5
}

{\displaystyle t\_{0,1} \{5\}}

## References

1. ↑ https://en.wikipedia.org/wiki/Decagon
2. ↑ http://www.polytope.net/hedrondude/polygons.htm
Conjoined Stars

**Conjoined stars** or **Siamese Stars** are hypothetical conjoined pair of stars that orbit so close they pull there mass to each other. There is much evidence supporting that stars like this exist. One such piece of evidence is that two stars in the extreme system VFTS 352 support the claim. If true, these two could be heading for a dramatic end, during which the two stars either merge together to create a single giant star, or form a binary black hole.
Dwarf planet

Pluto

A **dwarf planet** is a minor planet in a hydrostatic equilibrium orbiting some type of star or stellar remnant that is massive enough to be rounded by its own gravity, is not massive enough to cause thermonuclear fusion, and has not cleared its neighbouring region. Typically, they orbit a star in a Solar System within large belt that consists of other minor planets, as their gravity is not strong enough to clear its neighbourhood.

The most famous dwarf planet in our solar system is Pluto, as it was known as a planet for 76 years before it was demoted to dwarf planet. Well-known examples of other dwarf planets include Ceres in the asteroid belt, the ellipsoid-shaped Haumea, the cubewano Makemake, and the most massive dwarf planet Eris. There are likely to be hundreds of dwarf planets, and examples of strong candidates for being dwarf planets include Orcus, Salacia, Quaoar, Sedna, 2002 MS4, and Gonggong.

The majority of dwarf planets in our Solar system orbit in the Kuiper belt and Scattered disc, far past Neptune. This is likely the case because the gravity of the jovian planets flung them out into the distance, and the lack of material prevented them from growing any larger. There are exceptions, such as Ceres, which orbits in the Asteroid Belt and is classified as both an asteroid and a dwarf planet, and a new class of minor planet known as centaurs. Centaurs are minor planets whose orbits are between the outer planets' orbits. Centaurs are within the size range of a large asteroid to a small dwarf planet, and can have comet-like traits such as a coma. Their name comes from the fact that they are basically a hybrid between an asteroid and a comet, like how a centaur is a hybrid between a human and a horse. The largest centaurs are possibly large enough to be considered dwarf planets.

While it is certainly likely that other solar systems have dwarf planets, the candidates for dwarf planets, minor planets that haven't been verified to be dwarf planets, within our solar system are classed based on the placement of their orbit relative to planets.

## Gallery

Ceres, an asteroid and dwarf planetScale depictions of the largest dwarf planets and asteroids in our Solar System with colouration based on their colour indices and albedos.
Capacitor

A **capacitor** is an ideal circuit element that possesses only capacitance. The reciprocal of capacitance, elastance, is also associate with resistors; if this quantity is chosen, they may be called **elastors**.

## Capacitance

A current I, spread over a time t such that a charge Q has accumulated on the capacitor C experiences a voltage drop of

# V

Q
C

{\displaystyle V = \frac{Q}{C}}

.

Since the total charge accumulated is the integral of the current, this relation is also written as

# V

1
C

∫
I
d
t

{\displaystyle V = \frac{1}{C}\int I \text{d}t}

.

The capacitance of a capacitor is defined by this relation, and can be determined by the geometry of a capacitor through calculating the difference in potential between the two plates of the capacitor and then dividing this by the total charge on the plates.

### Parallel Plate Capacitor

A parallel plate capacitor has two flat plates held a distance apart. A positive charge accumulates on one plate, and a negative charge accumulates on the other plate.

The electric field E near a flat plate with surface charge density σ is

# E

σ

ε
0

{\displaystyle E = \frac{\sigma}{\varepsilon\_0}}

. Integrating between the two plates, at positions a and b along the r axis, with respect to distance gives the potential difference

# V

∫
a
b

σ

ϵ
0

d
# r

σ

ε
0

(

b
−
a

)

{\displaystyle V = \int\_a^b \frac{\sigma}{\epsilon\_0} \text{d}r = \frac{\sigma}{\varepsilon\_0} \left( b - a \right)}

.

Since

# C

Q
V

{\displaystyle C = \frac{Q}{V} }

, and the charge Q on the capacitor is equal to the charge density σ multiplied by the area Σ,

# C

σ
Σ

σ

ε
0

(

b
−
a

)

{\displaystyle C = \frac{\sigma\Sigma}{\frac{\sigma}{\varepsilon\_0} \left( b - a \right)}}

.

Simplifying gives the capacitance

# C

ε
0

Σ

b
−
a

{\displaystyle C = \varepsilon\_0 \frac{\Sigma}{b-a}}

### Cylindrical Capacitor

In a cylindrical capacitor, the charge is stored on two cylindrical plates, positioned at a radius a and b.

The electric field E around a cylindrical shell of radius a is

# E

σ
a

ε
0

r

{\displaystyle E = \frac{\sigma a }{\varepsilon\_0 r}}

. Integrating between the two cylinders, at positions a and b along the r axis, with respect to distance gives the potential difference

# V

∫
a
b

σ
a

ε
0

r

d
# r

σ
a

ε
0

ln
⁡

(

b
a

)

{\displaystyle V = \int\_a^b \frac{\sigma a }{\varepsilon\_0 r} \text{d}r = \frac{\sigma a}{\varepsilon\_0} \ln \left( \frac{b}{a} \right)}

.

Since

# C

Q
V

{\displaystyle C = \frac{Q}{V} }

, and the charge Q on the capacitor is equal to the charge density σ multiplied by the area

# Σ
2
π
a
L

{\displaystyle \Sigma = 2\pi a L}

,

# C

σ
2
π
a
L

σ
a

ε
0

ln
⁡

(

b
a

)

{\displaystyle C = \frac{\sigma 2\pi a L}{\frac{\sigma a}{\varepsilon\_0} \ln \left( \frac{b}{a} \right)}}

.

Simplifying gives the capacitance

# C

2
π

ε
0

L

ln
⁡

(

b
a

)

{\displaystyle C = \frac{2\pi\varepsilon\_0 L}{ \ln \left( \frac{b}{a} \right)}}

.
Circle multiverse

A **circle multiverse** is a type of 4+1-dimensional multiverse in which multiple universes are arranged in a ring, with each universe being adjacent to exactly two other universes (one in each direction). Traveling far enough in a circle multiverse will eventually take the traveler back to their origin universe. The rings shape does not need to be regular and its curvature does not need to be regular. It can even have straight segments or segments with negative curvature, as long as it comes back into itself at some point. Its shape can be anything but its topology must be that of a circle.

A similar type of multiverse, but without any curvature, is a line multiverse.
Cora

**Cora** is a nearly indestructible, somewhat fluid material that can be found on the outside of beyond bubbles. Cora is the material of which these bubbles are constructed of, so therefore, The Guardian Of The 5th Wall can create it easily.

Omnipotence^1 entities or beyond can pass, or simply touch the material, so normal people can't look other realities. This can be created or destroyed by any omnipotent^G64 entity or above, but entities like The Aspect Lords or The True God have no reason of creating it.
2 41 polytope

The **241 polytope**, also known as the **diacositetracont-myriaheptachiliadiacosioctaconta-zetton**, is a polyzetton based around the bifurcating Coxeter-Dynkin diagram with sequences of length 2, 4 and 1, with a single ringed node at the end of the sequence of length 2.

It has the Bowers acronym **bay** and the Bowers symbol x3o3o3o \*c3o3o3o3o.

## Structure and Sections

### Subfacets

* 2160 points (0D)
* 69120 line segments (1D)
* 483840 triangles (2D)
* 1209600 tetrahedra (3D)
* 1209600 pentachora (4D)
* 60480 pentacrosses (5D)
* 483840 hexatera (5D)
* 6720 icosiheptaheptacontidipeta (6D)
* 138240 heptapeta (6D)
* 240 231 polytopes (7D)
* 17280 octaexa (7D)
* 1 241 polytope (8D)
Altered disnub icosahedron

An **altered disnub icosahedron** is a uniform compound polyhedron. It is the compound of 20 octahedra, but is different from the disnub icosahedron, which has 60 vertices instead of 120.

Its Bowers acronym is **addasi**.

## Structure and Sections

### Subfacets

* 120 points (0D)
* 240 line segments (1D)
* 160 triangles (2D)
* 1 altered disnub icosahedron (3D)

## See Also

* Polytope Wiki. "Altered disnub icosahedron"