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Axiomatic Panbiogeography

offers an application of incidence geometry to historical biogeography by defining collection localities as points, tracks as lines and generalized tracks as planes.
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Track Analysis beyond Pan

From main massing differentiate concurrent polarity links


How do we differentiate polar concurrent opposites so as to divide out parallels from other-wise metrically identifiably unique patterns?



Masses provide the sign vehicles to accomplish this, ensuring that biogeometric parallels are not thought as the same thing as biased (directionally) polarity based distributions. The use of grossone density rather than fractal dimension permits this and can explain Kant's "density".  Width per parallel depends on density to volume scale.



So when  used between the baseline direction and node-antinode space the mass can impart a width to the track that can avoid the "inadvisability" suggested by Morrone (below).  The number of holes within the absorbing area relates the species density to panbiogeographic mass. Species space usage is thus a measure of this density per geologic "volume" (historical geography over time).



"That is, with JUNG, you can not discriminate the calculation to traverse particular paths. For those familiar with PageRank, you can’t arbitrarily bias the random walker (see Grammar-Based Random Walkers in Semantics Networks). The best that we can achieve with JUNG is to bias it with edge weights and filtered edge labels. In the example below, an EdgeLabelTransformer (from Blueprints) is used to only allow the traverser to follow knows edges in the graph. As such, besides the background alpha-probability, only vertices 2 and 4 get rank as they are the head of knows edges."

This may enable one to define a mass with respect to a density of nodes since edge shape (weight) could sort lat and long tracks per Z lexicographic ordering (dimensional reduction 2-d to 1-d).







Parallels in mobilism and immobilism are discriminated in distributions from polarized distributions themselves.

Parallel graphs where heterogenous page-ranked found masses across monphyla of limited geographic distribution may work with changing grossone numeral systems across the parallels.

An interesting aspect of panbiogeography off Earth is the possibility of predicting if diagrams of area homology indicate more than one phylogenetic trajectory from Outerspace. Once a full experimental use of panbiogeographic mass is integrated into specific instances of the Croizat Method it is possible that a weight effect of distributions within a gravitational field (possibly connected via quatum mechanical peculiarity to levels of organization (no matter the level of selection))(plants having a compressive force mostly vertical and animals utilizing the forms of energy via empty space traversed during form-making) may be analyzed by division of the geographic distributions into those from Mars vs Venus out of orbits vs "seeding" from even further out of the solar system space.
 
This possible research is one advantage that authentic panbiogeography offers the field of historical biogeography. There may come a day when axiomatic panbiogeography could predict whether there are only one kind of coming from off Earth distributions within the data of geographic distributions or whether if life were found on Mars, how that might be connected to different biotic area homologies on Earth.




 

How individual and general tracks may be dual projective geometries of a common mass - individual track relation:






 




 

This same question c an be asked for in the Salamander Genus Desmognathus(map data citation).