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The term triple torus is also occasionally used to denote a genus 3 surface. [7] [5] The Klein quartic is a compact Riemann surface of genus 3 with the highest possible order automorphism group for compact Riemann surfaces of genus 3. It has 168 orientation-preserving automorphisms, and 336 automorphisms altogether. Several genus 3 surfaces
The genus of a 3-dimensional handlebody is an integer representing the maximum number of cuttings along embedded disks without rendering the resultant manifold disconnected. It is equal to the number of handles on it. For instance: A ball has genus 0. A solid torus D 2 × S 1 has genus 1.
Poloidal direction (red arrow) and toroidal direction (blue arrow) A torus of revolution in 3-space can be parametrized as: [2] (,) = (+ ) (,) = (+ ) (,) = using angular coordinates θ, φ ∈ [0, 2π), representing rotation around the tube and rotation around the torus's axis of revolution, respectively, where the major radius R is the distance from the center of the tube to ...
The mapping torus M g of a homeomorphism g of a surface S is the 3-manifold obtained from S × [0,1] by gluing S × {0} to S × {1} using g. If S has genus at least two, the geometric structure of M g is related to the type of g in the classification as follows: If g is periodic, then M g has an H 2 × R structure;
All have a standard splitting of genus one. This is the image of the Clifford torus in under the quotient map used to define the lens space in question. It follows from the structure of the mapping class group of the two-torus that only lens spaces have splittings of genus one. Three-torus
This group is isomorphic to both PSL(2, 7) and PSL(3, 2). For genus 4, Bring's surface is a highly symmetric surface. For genus 7 the order is maximized by the Macbeath surface, with order 504; this is the second Hurwitz surface, and its automorphism group is isomorphic to PSL(2, 8), the fourth-smallest non-abelian simple group.
A polyhedral torus can be constructed to approximate a torus surface, from a net of quadrilateral faces, like this 6x4 example. In geometry, a toroidal polyhedron is a polyhedron which is also a toroid (a g-holed torus), having a topological genus (g) of 1 or greater. Notable examples include the Császár and Szilassi polyhedra.
To obtain an example of a Hurwitz group, let us start with a (2,3,7)-tiling of the hyperbolic plane. Its full symmetry group is the full (2,3,7) triangle group generated by the reflections across the sides of a single fundamental triangle with the angles π/2, π/3 and π/7. Since a reflection flips the triangle and changes the orientation, we ...