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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
Quaternion Algebraic Geom
Primate Vicariances
Individual Track Construc
Generalized Tracks
Main Massings
Track Analysis and MetaCo
Martitrack Panbiogeograph
Replies to Criticism
Multimodel Selection
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Track Analysis beyond Pan

Quaternion Algebraic Geometry: a means to transform collection locality data into taxanomic topologies and vice-versa as a function of tracks, nodes, masses and baselines.
 Head's (Molecular Panbiogeography of the Tropics)

page 5- 6 -- writes, " How did the distribution of a plant or animal develop?  One theory is that such a pattern originated by a plant or animal evolving at a point somewhere within its current area and spreading out from  there to the limits of its present range.  Researchers attempt to locate the "center of origin" or " ancestral area" by studying the distribution and phylogeny in the group itself and by using different criteria...Finding the center of origin of a group is a fundamental aim of many studies, and groups may be analyzed in ever-increasing detail in order to locate the center."

"An alternative approach considers a group not on its own, but in relationship to its closest relative or sister group (Fig 1-2)...Each of the two groups may have arisen not by spreading out from a point, but by geographic (allopatric) differentitation in its respective area from a widespread ancestor.  In this process ("vicariance")there is no physical movement, only differentiation, with populations in area A evolving into one form and populations in area B into another." (Heads page 7)

What I imagine here Wright's path analysis between the graphical presentation of sets of biquaternions.  Inside of that-- one can establish otherwise thought centers of origin which are split as the actual quaternionic groupings are reconstructed with empirically distributed data.


Here we show how to create coefficients (geometrically) of tracks nodes and masses to return (algebraically) the two groups (1-7) and (8-15). Approximate lats and longs will be assumed for the square areas. There is only one baseline here.

Each collection locality receives a coefficient for the individual track it is a part of as well as for any node it may be connected via its track to.These coefficients embody components of the three angles needed to trace the point to circle in the Southern Hemisphere (where algebraic geometry applies). A quick glance appears to present at least 3 or 5 nodes and a single mass.

Methods exist to "frame" 3D orientations of quaternions which may provide means to differentiate tracks from nodes from masses directly.


 Here I attempt to show how differential geometrical quaternionic manifolds can model vicarance and provide detailed "empty space" constructions of supposed centers of origin.  This permits ordination trends to be correlated with points, nodes, masses and baselines without need to use Aristotelian dichotomies but still provides a place where cladistic nodes and dichotomous key branch points could be organized relative to Croizat's method. Quaternionic algebra provides the means to relate the baselines to distributions and locate these places where other researches are searching for centers of origin (but there is really no movement there in the majority of cases) and using" trees".  Although I had first thought that the right and left action of quaternions may embody the ordinations (trends) to systematic classifications via biquaterions it may be possible that the entire Aristotleian dichotomy maybe completely forward figured in the math if the dual external form 1-D projections can be related to the panbiogeographic function through the algebra rather than having the algebra only relate finite an dinfininte baslines per mass. This would make torsion free properties apply at the level of masses rather than be only available for nodes and may work to recreate the systematic topology through translation and scaling (branch lengths) with geometric algebraic rotors.

During the construction of the matching between the collection localities, through modeled coefficients to support any given classification, each coefficient is chosen both to potentially match to a range of possible classifications and different sets of generalized tracks and baselines but does so by given each pair of collection localities chosen 3 different angles.  By adusting the pairs started with during the algorthim processing closer matches between panbiog modeles and taxonomies may be approached.

A Panbiogeographic use of Path Analysis

The “statistical” nature of Croizat’s method has been challenged ( Patterson refs).  Here Wright’s path analysis is applied to expanding the notion of the panbiogeographic node beyond being a simple junction/connection between tracks and a notion of track variability is applied to create a interpretative basis from which track graphs become the statistical graphs which on average per mass within baselines they have always been.  Wright’s idea of path coefficients clarifies the notion of causation captured in Croizat’s panbiogeographic method.


The spatial embedding of a track within a node within a mass within a baseline provides the one way causal/correlational directums that Wright described analogically with shadows and light.  (page 192 “The exception (with respect to reciprocal action) is where one variable is clearly external to the social system in question as is the influence of weather on crop yield)”  Thus simultaneous deviations can in those case be treated as lines of one way causation among the variables (track, node, mass, baseline).

The lack of a panbiogeographic spatial algorithm is due to no one connecting the gehnoritra of the Manual to the space, time and form of the Panbiogeographic synthesis along these this linearlity as expressed as generalized tracks. This analysis with path coeffiennts allows the shadow of taxonomic classifications to be realized as a temporal lag (phylogenetic- vicariant time) of reciprocally determined variables in common biogeographic – taxonomic circle of cause and effect.  Thus different polygenic taxa can indeed posses the same track and it is not the case as asserted by Parenti and Ebach that the spatial location can in no way inform the taxonomic classification.  This moving force is the inverse problem of using a given taxanomic correlation to guide in selection of the track node mass baseline path coefficients in the following sense:

Wright understood a recognizable notion of cause when he wrote in analogy in relativity that: “successions of events as involved in the movement of a shadow over a surface may indeed be reversed by change of viewpoint, if the shadow happens to be moving more rapidly than the velocity of light, but the  continuity of physical action here is not along the path of the shadow but traces separately to each point in this path from the points of interception of light.  There is frequently difficulty  in complex cases of distinguishing lines of direct causation from correlations due to common causation but in principle the distinction is clear enough.  Experimental intervention is possible only in the true lines of causation.

In the world of large scale events, certain patterns tend to recur. Certain recurrent successions of events come to be recognized, experimentally or otherwise, as lines of causation in the above sense.  Different lines of this character may come together in a certain type of event or may diverge from one.  In many cases a fairly adequate representation of the course of nature can obtained by viewing it as a coarse network in which “events” of interest are deviations in the values of certain measureable quantities.  A qualitative scheme depends on observation of sequences and experimental intervention.  It is of interst to make such a scheme at least roughly quantitative in the sense of evalutating the relative importance of action along different paths.  This was the primary purpose of the method of path coefficients.” (page `177)

Taxonomic trends of classification correlations are Wright’s shadows moving on the surface of life and earth evolving together (Croziat’s geographic distributions).  The Croizat method reduces in an effect to an assertion that  the shadow does not move faster than the points the light (tracks) map out!

Here one is determining the correlation of two individual tracks into a general track given 3 nodes spatial beyond the variation in track width.


It is still taking me some time to complete some kind of mechanical algorthim that implements an subjective aspect of Croizat's method in a more objective way than "by eye" but I think I have published all of the needed information for anyone else to do so.  With crickets, fish, bats and  salamanders in the same track I think I should have enough diversity data to ensure that the process (working (fingers crossed)) can not be dismissed as a specialist focus nevertheless some of the comments below are well taken. Vicariance is a part of the method but it is not the meaning of space (that is only the disjunction or separation (NO MATTER THE FRIABILITY)).  Cause and correlation must be seperated. It seems that even the use of grossone infinity if a proper path diagram were constructed could be use to refine the vicariance notion(adding random arbitary objects on top may be needed). I think I am finally settling on Structural Equation Modelling to embody the link between the 3 modeled angles per collection locality and the potentially infinite (per continuum) coefficients to Croizat concepts per linear point set.  When working with only salamanders I could not be as sure that I was not seperating cause and corrleation in the first approximation.

 Hi there,

I have to confess that these discussions really help me sorting my pdf 

ok, now... I agree with your reading about the means of dispersal, 
Croizat pretty much found no correlation between means of dispersal and 
distribution and this was major. I partially disagree, however, with 
your opinion that the primary problem is to find whether the history of 
earth and of life are correlated. The "earth and life evolve together" 
phrase is kind of a primary axiom of panbiogeography, so from the very 
beginning one assumes there is a correlation between the evolution of 
living forms and the geological landscape where they evolve. But I guess 
this a minor discrepancy...

> Well its worse than bad. The content is derived from publications that
> would not be out of your reach. A track is defined as a minimal spanning
> tree, where localities are linked by the shortest geographic distance IN
> THE ABSENCE OF OTHER INFORMATION. As I pointed out in an earlier
> posting, one can link individual taxa together first, and then link
> related taxa in the order of their relationship (or just their minimal
> distance).
I meant, I have not read the book... I have read the literature, though. 
I know what a track is and how you can draw one, and my point was 
precisely that since you can link the localities using a nearest 
neighbor criterion you don't need a phylogeny for the group you are 
working with.
> As I mentioned in an earlier posting, a beginner paper posted on my
> website that covers various approaches although I do not go into
> quantitative algorithms as its not my forte.
>> han/evolutionary-biography/panbiogeographic-publications/grehan-publicat
> ions/
>well, I have been trying for years now to get panbiogeography's 
algorithms. Aside from Page (1987) and Henderson (1989) I've had 
troubles finding any. That's why I used Track compatibility analysis as 
proposed by Craw. In doing this I've found that the method is not 
spatial at all, and its results are maximum vicariance. I've seen papers 
using PAE to find generalized tracks in an unexplained way: I still 
don't know how to turn a steiner tree into a minimun spanning tree; I 
think this is not possible at all, but there are published papers using 
PAE to derive generalized tracks... You also have, of course, the 
classic "by eye"...

 >Please cite the publications illustrating the frequent use of clique

Publications using track compatibility analysis...

Title Geographical diversification of tribes epilobieae, goncylocarpeae, 
and onagreae(onagraceae) in North America, based on parsimony analysis 
of endemicity and track compatibility analysis
Author(s) Katinas, L; Crisci, JV; Wagner, WL; Hoch, PC

Title Panbiogeography, biotic components and transition zones
Author(s) Morrone, Juan J.
Source Revista Brasileira de Entomologia 48 (2):149-162 2004

Title Historical biogeography of the Asteraceae from Tandilia and 
Ventania mountain ranges (Buenos Aires, Argentina)
Author(s) Crisci-V., Jorge; Freire-E., Susana; Sancho, Gisela; Katinas, 
Source Caldasia 23 (1):21-41 2001

Source JOURNAL OF BIOGEOGRAPHY 21 (1):97-109 1994

Author(s) MORRONE, JJ

Title Vicariance in historical biogeography: Analytical problems in 
reconstructing area cladograms
Author(s) Minaka, Nobuhiro
Source Acta Phytotaxonomica et Geobotanica 44 (2):151-184 1993

looking at the literature, it seems that PAE became more popular as a 
panbiogeographic technique at some point. Again, how you turn the 
results of PAE into a generalized tracks remains (at least to me) 

> No one's forcing you to read anything, but if one wants to critique
> panbiogeography (and there is certainly nothing wrong with that) it
> would be nice to see the critiques refer to specific applications that
> substantiate what may be perceived as problems.
The original email I send to Jason was not a critique to 
panbiogeography. I criticized the use of track compatibility for 
determining generalized tracks, the method is wrong even from a 
panbiogeographic perspective, and I gave specific applications and 
reasons why the method is wrong. I wrote "There is another problem with 
panbiogeography...", ok, sorry, please read: there is a problem with the 
application of track compatibility analysis for discovering generalized 

I also mentioned that because you can use a nearest neighbor criterion 
to draw the tracks panbiogeography was used as an escape when you want 
to do historical biogeography and don't have cladograms to do BPA for 
instance. I still think most people do this, at least in Latin America 
where you have many species and not too many cladograms or money to work 
on them...


Rather than trying to code the imagined panbiog algorithm to relate “species” and “gene” trees through the track graph map as begun above I am going to see how far one can accomplish the same goal using graph traversals rather than pattern matching.  In this mode it would be necessary to always have a graph traversal of a heterogeneous graph database that can go from any vertex through the panbiog Atlas for the monophyla under consideration and also hit all of terminal leaves in the component phylogeny (suggested from morphology or DNA etc) in the heterogeneous total graph.  Additionally the traversal though rooted in the track graph subset must be able to return the entire phylogeny as the graph is updated with new collection localities.

"Gremlin 1.1 now provides a table data structure and the ability to name steps in a path. These two features conveniently provide traversal-based graph pattern matching. Now it is possible to emulate the behavior of SPARQL in property graphs while simultaneously, leveraging the traversal aspects of Gremlin."

It will be  possible by "slicing an dicing" this table that the different gene vs Species trees can be compared via the track^node^mass^basline graphs.  Thus one will be able to reverse the position between Taxonomy and Historical Biogeography as Croizat often wrote it.  Rather than being the last but not least biogeograpic graph databases with a Panbiogeographic Atlas will be able help inform differences in proposed phylogenies. The node of the cladogram can be sustained but even genetic (as opposed to purely geological) vicariance can be objectified.

It seems that one may be able to do a pattern match between the phlogenetic/cladogenic trees (morphology vs DNA) and then do a traversal through the track graph adjusting it (via post slice and dice) to create a total graph that better fits the geographic data to on or the other monphyletic tree by looping back with "backtrack". There may be a memory storage limit as larger phylogenies are computed.

"Given the imperative nature of Gremlin, you will have to know how your data set looks — in other words, are there more ‘knows’ than ‘bought’ edges? In theory, you can get the same O(?) out of Gremlin as say, SPARQL over an RDF triple store—you just have to be smart about making sure you pick the smallest set of things to start from — and you always do start somewhere (even in pattern matching).

Hope that is clear,

This, as the edge is the "ground" (track^node^mass^baseline) we will know thanks to Croizat what the ontological divisions look like and how many tracks per node per mass per baseline exist in the Panbiogeographic Atlas per taxa.

Whether this background to the derivative at the node or the continuity in the area (Proposition 2.1) indicated that Gremlin  for an infinity computer with various [0,1) intervals per derivative needs be created first remains to be seen.

Although this is no excuse but the algorithm that relates the track^node^mass^baseline to the cladogram requires that the alpha beta and gamma  angles take the mutilrelational track into singlerelational space that is also set to contain the phylogenetic trees possible.

On the use of the Infinty Computer to create parallel processing in the Web of Data

The subject object predicate URI structure in the web of data could be processed in parallel by use of a trajectory from infinitesimal to infinite for each subject object pair across a grossone(1) to a grossone(2) numeral system in a complex graph.  Each subject and object has its own grossone representation with the finite gross digits binding the parallel between the (1) and (2) graphed storages.

The panbiogeographic track^node^mass^basline ontology per collection locality is an index already embedded in the geograhpic distribution data.
The Croizat method consists in part in extracting this index as it coordinates Space to Form-making through Time.

This should be possible with Ubigraph or some software developed from it.

Ubigraph could be used with an affine shell to sort out the edge nodes between Cottus and Aneides beyond what was suggested by a quick look only at outline maps and grading S- N Aneides flavipunctatus - Cottus gulosus
Aneides ferreus - Cottus perplexus  Aneides lugubris - Cottus asper Aneides vagrans - Cottus klamathensis