18:
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The 15 remaining ordinary double points at infinity correspond to the 15 lines that pass through the opposite vertices of the inscribed icosidodecahedron, all 15 of which also intersect in the center of the figure.
104:. To these 30 icosidodecahedral vertices are added the summit vertices of the 20 tetrahedral shapes. These 20 points themselves are the vertices of a concentric
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81:) showed that 65 is the maximum number of double points possible. The Barth sextic is a counterexample to an incorrect claim by
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The Barth Sextic may be visualized in three dimensions as featuring 50 finite and 15 infinite ordinary double points (nodes).
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circumscribed about the inner icosidodecahedron. Together, these are the 50 finite ordinary double points of the figure.
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shapes oriented such that the bases of these four-sided "outward pointing" shapes form the triangular faces of a regular
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Referring to the figure, the 50 finite ordinary double points are arrayed as the vertices of 20 roughly
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182:(1996), "Two projective surfaces with many nodes, admitting the symmetries of the icosahedron",
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Jaffe, David B.; Ruberman, Daniel (1997), "A sextic surface cannot have 66 nodes",
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Informal accounting of the 65 ordinary double points of the Barth Sextic
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in 1946 that 52 is the maximum number of double points possible.
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26:
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in 3 dimensions with large numbers of double points found by
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77:, David Jaffe and Daniel Ruberman (
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30:Real ordinary double points of the Barth Sextic.
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66:of degree 6 with 65 double points, and the
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25:
15:
291:
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70:of degree 10 with 345 double points.
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13:
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16:
278:"Animations of Barth surfaces"
1:
205:Journal of Algebraic Geometry
184:Journal of Algebraic Geometry
166:American Mathematical Society
146:
7:
119:
10:
320:
141:List of algebraic surfaces
73:For degree 6 surfaces in
62:). Two examples are the
22:3D model of Barth-sextic
50:is one of the complex
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31:
23:
37:
29:
21:
106:regular dodecahedron
299:Algebraic surfaces
256:Weisstein, Eric W.
156:(April 15, 2016),
44:algebraic geometry
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32:
24:
136:Togliatti surface
102:icosidodecahedron
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304:Complex surfaces
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280:. Archived from
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246:. Archived from
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234:. Archived from
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83:Francesco Severi
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126:Endrass surface
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287:
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284:on 2008-01-25.
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250:on 2012-02-19.
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238:on 2012-02-19.
232:"Barth sextic"
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225:External links
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211:(1): 151–168,
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190:(1): 173–186,
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162:Visual Insight
158:"Barth Sextic"
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56:Wolf Barth
52:nodal surfaces
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244:"Barth decic"
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48:Barth surface
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282:the original
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260:Barth Sextic
248:the original
236:the original
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169:, retrieved
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64:Barth sextic
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264:Barth Decic
98:tetrahedral
68:Barth decic
38:Barth Decic
293:Categories
171:2016-12-27
154:Baez, John
147:References
269:MathWorld
180:Barth, W.
114:Baez 2016
120:See also
217:1486992
196:1358040
58: (
266:") at
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262:" ("
79:1997
60:1996
46:, a
258:, "
116:).
42:In
295::
213:MR
207:,
192:MR
186:,
164:,
160:,
272:.
220:.
209:6
199:.
188:5
175:.
112:(
75:P
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