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- May 10, 2019 for Mobius Maxi, with firmware V2.00 or above. Preview from your phone in real time; View and download your photos and videos taken by Mobius Cam, right on your phone; To set your Mobius Cam's parameters easily.
- Mobius v1.2 is a file encryption, decryption program for Windows. Here are just a few of the features: 100% security against encryption cracking, lightning fast speed, easy to use help files, such.
- Ecotech Mobius App
- Mobius Action Camera App
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- Mobius Clinic App
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Download Mobius USB Tools apk 2.0.1.4 for Android. Unpublished due to death threats.
In geometry, a Schwarz triangle, named after Hermann Schwarz, is a spherical triangle that can be used to tile a sphere, possibly overlapping, through reflections in its edges. They were classified in (Schwarz 1873).
These can be defined more generally as tessellations of the sphere, the Euclidean plane, or the hyperbolic plane. Each Schwarz triangle on a sphere defines a finite group, while on the Euclidean or hyperbolic plane they define an infinite group.
A Schwarz triangle is represented by three rational numbers (pqr) each representing the angle at a vertex. The value n/d means the vertex angle is d/n of the half-circle. '2' means a right triangle. When these are whole numbers, the triangle is called a Möbius triangle, and corresponds to a non-overlapping tiling, and the symmetry group is called a triangle group. In the sphere there are three Möbius triangles plus one one-parameter family; in the plane there are three Möbius triangles, while in hyperbolic space there is a three-parameter family of Möbius triangles, and no exceptional objects.
Solution space[edit]
A fundamental domain triangle (pqr), with vertex angles π/p, π/q, and π/r, can exist in different spaces depending on the value of the sum of the reciprocals of these integers:
This is simply a way of saying that in Euclidean space the interior angles of a triangle sum to π, while on a sphere they sum to an angle greater than π, and on hyperbolic space they sum to less.
Graphical representation[edit]
A Schwarz triangle is represented graphically by a triangular graph. Each node represents an edge (mirror) of the Schwarz triangle. Each edge is labeled by a rational value corresponding to the reflection order, being π/vertex angle.
Schwarz triangle (pqr) on sphere | Schwarz triangle graph |
Order-2 edges represent perpendicular mirrors that can be ignored in this diagram. The Coxeter-Dynkin diagram represents this triangular graph with order-2 edges hidden.
A Coxeter group can be used for a simpler notation, as (pqr) for cyclic graphs, and (pq 2) = [p,q] for (right triangles), and (p 2 2) = [p]×[].
A list of Schwarz triangles[edit]
Möbius triangles for the sphere[edit]
(2 2 2) or [2,2] | (3 2 2) or [3,2] | ... |
---|---|---|
(3 3 2) or [3,3] | (4 3 2) or [4,3] | (5 3 2) or [5,3] |
Schwarz triangles with whole numbers, also called Möbius triangles, include one 1-parameter family and three exceptional cases:
- [p,2] or (p 2 2) – Dihedral symmetry,
- [3,3] or (3 3 2) – Tetrahedral symmetry,
- [4,3] or (4 3 2) – Octahedral symmetry,
- [5,3] or (5 3 2) – Icosahedral symmetry,
Schwarz triangles for the sphere by density[edit]
The Schwarz triangles (pqr), grouped by density:
Density | Dihedral | Tetrahedral | Octahedral | Icosahedral |
---|---|---|---|---|
d | (2 2 n/d) | |||
1 | (2 3 3) | (2 3 4) | (2 3 5) | |
2 | (3/2 3 3) | (3/2 4 4) | (3/2 5 5), (5/2 3 3) | |
3 | (2 3/2 3) | (2 5/2 5) | ||
4 | (3 4/3 4) | (3 5/3 5) | ||
5 | (2 3/2 3/2) | (2 3/2 4) | ||
6 | (3/2 3/2 3/2) | (5/2 5/2 5/2), (3/2 3 5), (5/4 5 5) | ||
7 | (2 3 4/3) | (2 3 5/2) | ||
8 | (3/2 5/2 5) | |||
9 | (2 5/3 5) | |||
10 | (3 5/3 5/2), (3 5/4 5) | |||
11 | (2 3/2 4/3) | (2 3/2 5) | ||
13 | (2 3 5/3) | |||
14 | (3/2 4/3 4/3) | (3/2 5/2 5/2), (3 3 5/4) | ||
16 | (3 5/4 5/2) | |||
17 | (2 3/2 5/2) | |||
18 | (3/2 3 5/3), (5/3 5/3 5/2) | |||
19 | (2 3 5/4) | |||
21 | (2 5/4 5/2) | |||
22 | (3/2 3/2 5/2) | |||
23 | (2 3/2 5/3) | |||
26 | (3/2 5/3 5/3) | |||
27 | (2 5/4 5/3) | |||
29 | (2 3/2 5/4) | |||
32 | (3/2 5/4 5/3) | |||
34 | (3/2 3/2 5/4) | |||
38 | (3/2 5/4 5/4) | |||
42 | (5/4 5/4 5/4) |
Triangles for the Euclidean plane[edit]
(3 3 3) | (4 4 2) | (6 3 2) |
Density 1:
- (3 3 3) – 60-60-60 (equilateral),
- (4 4 2) – 45-45-90 (isosceles right),
- (6 3 2) – 30-60-90,
Density 2:
- (6 6 3/2) - 120-30-30 triangle
Density ∞:
- (4 4/3 ∞)
- (3 3/2 ∞)
- (6 6/5 ∞)
Triangles for the hyperbolic plane[edit]
(7 3 2) | (8 3 2) | (5 4 2) |
(4 3 3) | (4 4 3) | (∞ ∞ ∞) |
Fundamental domains of (pqr) triangles |
Density 1:
- (2 3 7), (2 3 8), (2 3 9) ... (2 3 ∞)
- (2 4 5), (2 4 6), (2 4 7) ... (2 4 ∞)
- (2 5 5), (2 5 6), (2 5 7) ... (2 5 ∞)
- (2 6 6), (2 6 7), (2 6 8) ... (2 6 ∞)
- (3 3 4), (3 3 5), (3 3 6) ... (3 3 ∞)
- (3 4 4), (3 4 5), (3 4 6) ... (3 4 ∞)
- (3 5 5), (3 5 6), (3 5 7) ... (3 5 ∞)
- (3 6 6), (3 6 7), (3 6 8) ... (3 6 ∞)
- ...
- (∞ ∞ ∞)
Density 2:
- (3/2 7 7), (3/2 8 8), (3/2 9 9) ... (3/2 ∞ ∞)
- (5/2 4 4), (5/2 5 5), (5/2 6 6) ... (5/2 ∞ ∞)
- (7/2 3 3), (7/2 4 4), (7/2 5 5) ... (7/2 ∞ ∞)
- (9/2 3 3), (9/2 4 4), (9/2 5 5) ... (9/2 ∞ ∞)
- ...
Density 3:
- (2 7/2 7), (2 9/2 9), (2 11/2 11) ...
Density 4:
- (7/3 3 7), (8/3 3 8), (3 10/3 10), (3 11/3 11) ...
Density 6:
- (7/4 7 7), (9/4 9 9), (11/4 11 11) ...
- (7/2 7/2 7/2), (9/2 9/2 9/2), ...
Density 10:
- (3 7/2 7)
The (2 3 7) Schwarz triangle is the smallest hyperbolic Schwarz triangle, and as such is of particular interest. Its triangle group (or more precisely the index 2 von Dyck group of orientation-preserving isometries) is the (2,3,7) triangle group, which is the universal group for all Hurwitz groups – maximal groups of isometries of Riemann surfaces. All Hurwitz groups are quotients of the (2,3,7) triangle group, and all Hurwitz surfaces are tiled by the (2,3,7) Schwarz triangle. The smallest Hurwitz group is the simple group of order 168, the second smallest non-abelian simple group, which is isomorphic to PSL(2,7), and the associated Hurwitz surface (of genus 3) is the Klein quartic.
The (2 3 8) triangle tiles the Bolza surface, a highly symmetric (but not Hurwitz) surface of genus 2.
The triangles with one noninteger angle, listed above, were first classified by Anthony W. Knapp in.[1] A list of triangles with multiple noninteger angles is given in.[2]
See also[edit]
Ecotech Mobius App
References[edit]
- ^A. W. Knapp, Doubly generated Fuchsian groups, Michigan Mathematics Journal 15 (1968), no. 3, 289–304
- ^Klimenko and Sakuma, Two-generator discrete subgroups of Isom( H 2 ) containing orientation-reversing elements, Geometriae DedicataOctober 1998, Volume 72, Issue 3, pp 247-282
- Coxeter, H.S.M. (1973), Regular Polytopes (Third ed.), Dover Publications, ISBN0-486-61480-8, Table 3: Schwarz's Triangles
- Magnus, Wilhelm (1974), Noneuclidean Tesselations and Their Groups, Academic Press, ISBN0080873774
- Schwarz, H. A. (1873), 'Ueber diejenigen Fälle in welchen die Gaussichen hypergeometrische Reihe eine algebraische Function ihres vierten Elementes darstellt', Journal für die reine und angewandte Mathematik, 75: 292–335, doi:10.1515/crll.1873.75.292, ISSN0075-4102 (Note that Coxeter references this as 'Zur Theorie der hypergeometrischen Reihe', which is the short title used in the journal page headers).
- Wenninger, Magnus J. (1979), 'An introduction to the notion of polyhedral density', Spherical models, CUP Archive, pp. 132–134, ISBN978-0-521-22279-2
External links[edit]
Mobius Action Camera App
- Weisstein, Eric W.'Schwarz triangle'. MathWorld.
- Klitzing, Richard. '3D The general Schwarz triangle (p q r) and the generalized incidence matrices of the corresponding polyhedra'.
Retrieved from 'https://en.wikipedia.org/w/index.php?title=Schwarz_triangle&oldid=941882452#Mobius_triangles'
## The Mobius Forms AppThis **Mobius Forms** App is an add-on to DNN. It is _the most customizable Form extension_ in the DNN ecosystem. This means that it- can be used to create a simple contact form in one minute
- can be modified to be any other form you need
- can be used as a starting point for your own AJAX forms in DNN
The app is built with the pattern Don't be DAFT (DAFT = Densely Abstract Features for Techies), aka the Anti-Abstraction pattern. So customizing it is mostly done using common technologies like HTML and if necessary, jQuery and C#.
Quick Intro To The Mobius Forms App for DNN
A DNN App is like a DNN module, just way better :). Since this is an open-code/open-source 2sxc-app, you can customize it to be anything you want! This list just shows what it already does, so you know what you get out-of-the-first-box.
- Pre-Built Forms for use or learning
- Basic contact form with Subject, Message, Name, E-Mail
- A support-request form with a dropdown-example
- An example with JS show/hide logic and saving raw JSON-data
- AJAX, so no page reloads for validation, sending or messages
- Recaptcha (optional) validation on client and server
- data is saved, together with the Timestamp, SenderIP, optionaly generated Title or even raw JSON-data
- sends various e-mails, which are razor-templateable and has Reply-To and CC options
- multi-language labels and messages, already translated into English and German/Deutsch
- field validation uses html5 and jQueryValidation and works with multiple forms on the same page
- you can easily review / manage / filter the submitted items in a table-view
- export all submissions into an Excel-compatible XML format
- open code C# WebApi easy to customize if you ever need to
Mac App With Mobius Triangle Symbol
Because it's so simple and uses 2sxc, you can easily
- send more e-mails, trigger other custom WebApi actions
- create more custom forms which store into further content-types
Getting Started
Mobius Clinic App
This app is only useful is you use DNN. So assuming you have a DNN installation, all you need to do is install 2sxc and this app.
- Here's how to install 2sxc and an App of your Choice
- Now you can use this app as-is, or customize it to be whatever you need it to be.
- It probably helps to review the Overview about how the parts play together by default, so you can then change as little as necessary to get it to do what you want
Credits
Mac App With Mobius Triangle 1
The icon was built using the CC icon optical illusion by pedro baños cancer from the Noun Project