The degree 7 Klein graph and associated map embedded in an orientable surface of genus 3. The map has 24 vertices, 84 edges and 56 triangles.  This illustration is based on Klein's original drawing of the orbit of a fundamental region under the group of the Klein quartic [1]. The point labelled "p" appears seven times on the border. These are all identified, joining the seven triangles incident to "p". The point labelled "q" also appears seven times on the border. When these are identified, fourteen half triangles are joined. Both "p" and "q" are at maximal distance 3 from the center vertex and from each other. The remaining labels indicate edges that are joined, thereby matching the remaining fourteen half triangles. To assist in the identification of half triangles, a face coloring method suggested by MacKenzie is used [2].
<p>1. Klein, F. (1878). "Ueber die Transformation siebenter Ordnung der elliptischen Functionen" [On the order-seven transformation of elliptic functions]. Mathematische Annalen. 14 (3): 428–471. doi:10.1007/BF01677143. Translated in Levy, Silvio, ed. (1999). The Eightfold Way. Cambridge University Press. <a href="//commons.wikimedia.org/wiki/Special:BookSources/978-0-521-66066-2" title="Special:BookSources/978-0-521-66066-2">ISBN 978-0-521-66066-2</a>. MR 1722410.
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2. Dana Mackenzie (1995) A Hyperbolic Plane Coloring and the Simple Group of Order 168, The American Mathematical Monthly, 102:8, 706-715, DOI: 10.1080/00029890.1995.12004646
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