Circle (EntityTopic, 15)
From Hi.gher. Space
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| + | <[#ontology [kind topic] [cats 2D Hypersphere] [alt [[freebase:01vkl]] [[wikipedia:Circle]]]]> | ||
{{STS Shape | {{STS Shape | ||
| dim=2 | | dim=2 | ||
| - | | elements=1, 0 | + | | elements=1 circular edge, 0 |
| genus=0 | | genus=0 | ||
| ssc=(xy) | | ssc=(xy) | ||
| + | | ssc2=T2 | ||
| pv_circle=1 | | pv_circle=1 | ||
| pv_square=<sup>π</sup>⁄<sub>4</sub> ≈ 0.7854 | | pv_square=<sup>π</sup>⁄<sub>4</sub> ≈ 0.7854 | ||
| - | | extra={{STS | + | | extra={{STS Matrix| |
| - | | | + | 0 0 |
| - | | notation=2 ( | + | 2 1}}{{STS Tapertope |
| - | | index= | + | | order=1, 0 |
| + | | notation=2 | ||
| + | | index=2 | ||
| + | }}{{STS Toratope | ||
| + | | expand=[[Circle|2]] | ||
| + | | notation=(II) | ||
| + | | index=1b | ||
}}{{STS Bracketope | }}{{STS Bracketope | ||
| - | | index= | + | | index=3 |
| + | | notation=(II) | ||
}}}} | }}}} | ||
A '''circle''' refers to the surface of a perfectly symmetrical [[planar]] object. | A '''circle''' refers to the surface of a perfectly symmetrical [[planar]] object. | ||
| + | |||
| + | Sometimes, the word circle is used to refer to the edge, while the filled in version is called a ''disc''. | ||
== Equations == | == Equations == | ||
| Line 28: | Line 39: | ||
*The [[linear]] [[cross-section]]s (''n'') of a circle are: | *The [[linear]] [[cross-section]]s (''n'') of a circle are: | ||
| - | <blockquote>[!x,!y] ⇒ [[ | + | <blockquote>[!x,!y] ⇒ [[digon]] of length (''r''cos(π''n''/2))</blockquote> |
| + | |||
| + | == Homology groups == | ||
| + | Any unstated homology group is the trivial group 0. | ||
| + | |||
| + | ;0-frame:N/A | ||
| + | ;1-frame (circle):H<sub>0</sub> = ℤ, H<sub>1</sub> = ℤ | ||
| + | ;2-frame (disc):H<sub>0</sub> = ℤ | ||
== Use == | == Use == | ||
| Line 36: | Line 54: | ||
{{Dishapes}} | {{Dishapes}} | ||
| - | {{ | + | {{Tapertope Nav|1|2|3|1<br>Digon|2<br>Circle|11<br>Square|gons}} |
| - | {{Bracketope Nav|3|4| | + | {{Toratope Nav B||1|2|||II<br>Square|(II)<br>Circle|III<br>Cube|(III)<br>Sphere|gons}} |
| + | {{Bracketope Nav|2|3|4|[II]<br>Square|(II)<br>Circle|[III]<br>Cube|gons}} | ||
Latest revision as of 16:21, 25 March 2017
A circle refers to the surface of a perfectly symmetrical planar object.
Sometimes, the word circle is used to refer to the edge, while the filled in version is called a disc.
Equations
- Variables:
r ⇒ radius of circle
- All points (x, y) that lie on the surface of a circle will satisfy the following equation:
x2 + y2 = r2
- The hypervolumes of a circle are given by:
total edge length = 2πr
area = πr2
- The linear cross-sections (n) of a circle are:
[!x,!y] ⇒ digon of length (rcos(πn/2))
Homology groups
Any unstated homology group is the trivial group 0.
- 0-frame
- N/A
- 1-frame (circle)
- H0 = ℤ, H1 = ℤ
- 2-frame (disc)
- H0 = ℤ
Use
Circular faces are found in these trishapes on FGwiki:
| Notable Dishapes | |
| Flat: | triangle • square • pentagon • hexagon • octagon • decagon |
| Curved: | circle |
| 1. 1 Digon | 2. 2 Circle | 3. 11 Square |
| List of tapertopes | ||
| a. | b. | 1a. II Square | 1b. (II) Circle | 2a. III Cube | 2b. (III) Sphere |
| List of toratopes | |||||
| 2. [II] Square | 3. (II) Circle | 4. [III] Cube |
| List of bracketopes | ||
