Editing Triangle (section)

== Miscellaneous triangles ==
=== Circular triangles ===
{{main article|Circular triangle}}
[[File:Circular_triangles_convex_concave.png|thumb|upright=1.35|Circular triangles with a mixture of convex and concave edges]]
A [[circular triangle]] is a triangle with circular [[Arc (geometry)|arc]] edges. The edges of a circular triangle may be either convex (bending outward) or concave (bending inward).{{efn|1=A subset of a plane is [[convex set|convex]] if, given any two points in that subset, the whole line segment joining them also lies within that subset.}} The intersection of three [[Disk (mathematics)|disk]]s forms a circular triangle whose sides are all convex. An example of a circular triangle with three convex edges is a [[Reuleaux triangle]], which can be made by intersecting three circles of equal size. The construction may be performed with a compass alone without needing a straightedge, by the [[Mohr–Mascheroni theorem]]. Alternatively, it can be constructed by rounding the sides of an equilateral triangle.<ref>{{multiref
|{{harvnb|Hann|2014|p=[https://books.google.com/books?id=-CX-AgAAQBAJ&pg=PA34 34]}}
|{{harvnb|Hungerbühler|1994}}
}}</ref>

A special case of concave circular triangle can be seen in a [[pseudotriangle]].{{sfn|Vahedi|van der Stappen|2008|p=[https://books.google.com/books?id=SLo6okq4wVgC&pg=PA73 73]}} A pseudotriangle is a [[Simply connected space|simply-connected]] subset of the plane lying between three mutually tangent convex regions. These sides are three smoothed curved lines connecting their endpoints called the ''cusp points''. Any pseudotriangle can be partitioned into many pseudotriangles with the boundaries of convex disks and [[Bitangent|bitangent lines]], a process known as pseudo-triangulation. For <math> n </math> disks in a pseudotriangle, the partition gives <math> 2n - 2 </math> pseudotriangles and <math> 3n - 3 </math> bitangent lines.{{sfn|Pocchiola|Vegter|1999|p=[https://books.google.com/books?id=vtkaCAAAQBAJ&pg=PA259 259]}} The [[convex hull]] of any pseudotriangle is a triangle.{{sfn|Devadoss|O'Rourke|2011|p=[https://books.google.com/books?id=InJL6iAaIQQC&pg=PA93 93]}}

=== Triangle in non-planar space ===
{{main article|Hyperbolic triangle|Spherical triangle}}
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A non-planar triangle is a triangle not embedded in a [[Euclidean space]], roughly speaking a flat space. This means triangles may also be discovered in several spaces, as in [[hyperbolic space]] and [[spherical geometry]]. A triangle in hyperbolic space is called a [[hyperbolic triangle]], and it can be obtained by drawing on a negatively curved surface, such as a [[saddle surface]]. Likewise, a triangle in spherical geometry is called a [[spherical triangle]], and it can be obtained by drawing on a positively curved surface such as a [[sphere]].{{sfn|Nielsen|2021|p=[https://books.google.com/books?id=hHMjEAAAQBAJ&pg=PA154 154]}}

The triangles in both spaces have properties different from the triangles in Euclidean space. For example, as mentioned above, the internal angles of a triangle in Euclidean space always add up to 180°. However, the sum of the internal angles of a hyperbolic triangle is less than 180°, and for any spherical triangle, the sum is more than 180°.{{sfn|Nielsen|2021|p=[https://books.google.com/books?id=hHMjEAAAQBAJ&pg=PA154 154]}} In particular, it is possible to draw a triangle on a sphere such that the measure of each of its internal angles equals 90°, adding up to a total of 270°. By [[Girard's theorem]], the sum of the angles of a triangle on a sphere is <math> 180^\circ \times (1 + 4f) </math>, where <math> f </math> is the fraction of the sphere's area enclosed by the triangle.<ref>{{cite web|last=Polking |first=John C. |url=https://www.math.csi.cuny.edu/~ikofman/Polking/gos4.html |title=The area of a spherical triangle. Girard's Theorem. |website=Geometry of the Sphere |access-date=2024-08-19 |date=1999-04-25}}</ref><ref>{{cite web|last=Wood |first=John |url=https://homepages.math.uic.edu/~jwood/freshsem/ |title= LAS 100 — Freshman Seminar — Fall 1996: Reasoning with shape and quantity |access-date=2024-08-19}}</ref>

In more general spaces, there are [[comparison theorem]]s relating the properties of a triangle in the space to properties of a corresponding triangle in a model space like hyperbolic or elliptic space.{{sfn|Berger|2002|pp=134–139}} For example, a [[CAT(k) space]] is characterized by such comparisons.{{sfn|Ballmann|1995|p=viii+112}}

=== Fractal geometry ===
[[Fractal]] shapes based on triangles include the [[Sierpiński triangle|Sierpiński gasket]] and the [[Koch snowflake]].<ref>{{Cite book |last1=Frame |first1=Michael |url=https://books.google.com/books?id=i2axEAAAQBAJ&dq=koch+sierpinski&pg=PA21 |title=Fractal Worlds: Grown, Built, and Imagined |last2=Urry |first2=Amelia |date=2016-06-21 |publisher=Yale University Press |isbn=978-0-300-22070-4 |pages=21 |language=en}}</ref>