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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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Incidence Geometry
Composite Construction
Orthogenesis
Quaternion Algebraic Geom
Vicariance
Primate Vicariances
Individual Track Construc
Generalized Tracks
Planar Chaos
Graph Rewriting
Taxogeny
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Track Analysis and MetaCo
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Track Analysis beyond Pan

Generalized Tracks










The whole combination of gene trees and species trees depends on deciding which is on the right and which on the left.


On the Panbiogeographic track as Bezier Curve.



The Bezier curve is an interpolation curve methodology that seems  prima facie to capture significant aspects of the Panbiogeo

graphic track.  Patterson raised an issue with Croizat’s track by questioning whether the notion itself could be discriminated from being instead of the eminent graphs they were subsequently  argued to be (Page 1987) are rather  simple statistical artifacts of drawing connecting lines amongst large numbers of widely placed points.  Patternson’s idea is that simple by cherry picking points amongst a large set the appearance of order in the data can manufactured by human error rather than by distillation of significant average normalized behaviors.




The Bezier curve has the basic Croizat property to connect two different points (having the starting and ending points) but it also has the aspect of weighting the curve inside the convex hull of all control points near to each point on average””. 




The trick will be to pick the order of the points relative to the vicariance. The construction of the Bezier curve with Bernstein polynomials permits a definition of vicariance along the simple graph representation of a track and nodes (line connecting different collection localities). The vicariance is obtained at the midpoints.



The use of this curve results in the division of collection localities as those that are “control points” and those that are distribution points due to mobilism.




If vicariance in immobilism is presented by the relation of the polynomials to the track (any fracture of the track is equivalent despite the ordering of the points).  This relation is developed as one that connects the track width, node shape, mass density and baseline volume to the remaining non-control points and can further be used to link the baselined track graph to a phylogenetic trees proposed across the biogeography.

By altering the order (and hence affecting stability) one can have different edges responsible for different polynomial order vicariance.  Thus different kinds of vicariance can be explored for a set of points and compared to other sister groups. Alternatively the use of specific track widths, node shapes, mass densities and baseline models could suggest which amongst various possible are sisters.  The scalablity and translation and rotation of the curves enable one to search for sisters with complex embedded vicariance distributed across lineages.













 

 When the generalized track is formed through a mass rather an purely within nodal synthesis an.d older age is possible. Doubted possible principles of formation from Strecke parts (given the Oth track presented) can be orthogenically constructed rather than stipulated as an existence principle.  This requires a decision on the node to mass relation and gives a finite vs a larger infinite number of basline representations. That was not done in the use of South American generalized tracks though "vicariant" nodes above.




Two ants and two salamanders



Chorogaster (fish Amblydopsidae)CYAN Stereochilus ( salamander Desmognathinae) BLUE