Two examples of bounded complexes

Distance data between algae

This example is the bounded subcomplex of the unbounded polyhedron corresponding to some distance data between algae:

[1] 'Tobacco'              0.0 0.026086956 0.029347826 0.11521739 0.09130435 0.14456522 0.122826084 0.14130434
[2] 'Rice'                 0.027173912 0.0 0.041304346 0.12391304 0.10108696 0.1521739 0.13152175 0.14565217
[3] 'Marchantia'           0.030434782 0.041304346 0.0 0.10217391 0.07717391 0.13152175 0.12065218 0.13369565
[4] 'Chlamydomonas'        0.11304348 0.12065218 0.098913044 0.0 0.11195652 0.15 0.14239131 0.15652174
[5] 'Chlorella'            0.07826087 0.08695652 0.063043475 0.10108696 0.0 0.116304345 0.116304345 0.11521739
[6] 'Euglena'              0.13586956 0.14239131 0.12173913 0.14347826 0.12065218 0.0 0.1576087 0.13478261
[7] 'Anacystis_nidulans'   0.12391304 0.13152175 0.12065218 0.14565217 0.13043478 0.1673913 0.0 0.13695653
[8] 'Olithodiscus'         0.14130434 0.14456522 0.1326087 0.15869565 0.12826087 0.14347826 0.13586956 0.0

The colors in the picture encode the dimension of each cell:

green edges are 1-dimensional components of the bounded complex,

blue ones 2-dimensional,

red ones 3-dimensional, and

the purple edges in the center bound a 4-dimensional cell.

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An example from tropical geometry

This example (due to Bernd Surmfels) represents the incidence structure between points and lines in the Fano plane.
This point configuration P has tropical rank 2, but Kapranov rank 3; therefore it represents a tropical 2-polytope whose linear hull has dimension 3.

Here you can find a polymake file that represents the polytope according to Theorem 29 in [Develin & Sturmfels]; is the corresponding bounded subcomplex.

1 1 0 1 0 0 0
0 1 1 0 1 0 0
0 0 1 1 0 1 0
0 0 0 1 1 0 1
1 0 0 0 1 1 0
0 1 0 0 0 1 1
1 0 1 0 0 0 1

The complex has f-vector (15, 35, 21), and is a cone over the incidence graph of the Fano plane.

This is the (graph of the) complex. The 21 triangles are all incident to the central vertex.

Last modified: Tue Nov 4 10:02:54 PST 2003
julian@msri.org