File:Fano 3-space rectilinear.png
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![File:Fano 3-space rectilinear.png](https://upload.wikimedia.org/wikipedia/commons/thumb/2/2c/Fano_3-space_rectilinear.png/800px-Fano_3-space_rectilinear.png?20190718155037)
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Fano 3-space; rectilinear tetrahedral model thereof.
Summary
[edit]DescriptionFano 3-space rectilinear.png |
English: A purely rectilinear, tetrahedral model of the Fano 3-space, PG(3,2), which has 15 points, 35 lines, and 15 planes. Points A, B, C are at infinity (and so are points K, L, M, and O). Line AB may be thought of as being a horizon. Points D, E, H, F, G, I, N, J are the vertices of a parallelepiped seen in perspective. Triangles HFD and NJG are subtended by a pencil of lines issuing from C, therefore by the Desargues Theorem the intersection of DH and GN (K) is collinear with the intersections of HF and NJ (A), and GJ and DF (B). (This is just one example of the application of the Desargues Theorem in this diagram.) The colored line segments join the edge-points of Fano subplanes; they would be expected to be circular or elliptical or curved in a non-purely-rectilinear model. Points labeled with primed labels are meant to be identified with their corresponding non-prime-labeled points. Fano subplanes: (1) AKBMCLO, (2) AEDGCLI, (3) AEDFBKH, (4) BFDGCMJ, (5) BHEICMN, (6) AHFJCLN, (7) AIGJBKN, (8) KHDGCON, (9) MJDEAON, (10) LIDFBON, (11) DHKLMJI, (12) AIGFMOH, (13) BHEGLOJ, (14) CJFEKOI, (15) MLKEFGN. There are two possible 3D interpretations. One is that EDFHIGJN is a parallelepiped and points A, B, C are vanishing points, so the whole thing is seen in perspective. The other interpretation is that it is a projection (possibly orthographic) of a tetrahedron ABCD whose dimensions are entirely bounded. |
Date | |
Source | Own work |
Author | AugPi |
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[edit]I, the copyright holder of this work, hereby publish it under the following license:
Public domainPublic domainfalsefalse |
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I, the copyright holder of this work, release this work into the public domain. This applies worldwide. In some countries this may not be legally possible; if so: I grant anyone the right to use this work for any purpose, without any conditions, unless such conditions are required by law. |
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Date/Time | Thumbnail | Dimensions | User | Comment | |
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current | 15:50, 18 July 2019 | ![]() | 1,864 × 1,258 (313 KB) | AugPi (talk | contribs) | Changing the coloring of the colored line segments so that the meaning of the colors would be more logical. |
14:04, 17 July 2019 | ![]() | 1,852 × 1,282 (316 KB) | AugPi (talk | contribs) | User created page with UploadWizard |
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Horizontal resolution | 56.69 dpc |
Vertical resolution | 56.69 dpc |
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17 July 2019
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320,769 byte
1,258 pixel
1,864 pixel
39174ea46b7a7b56492f48dd85009c11152ebff7
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