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Galton's Law Revisited

Mendelian Morphometry

Ralph Abraham attempted to construct concepts bearing on the subject of Rene Thom’s morphogenesis (1972) in part in which he defined morphometry as a scheme for parameterizing observations with mathematical structures.  One of the objects of genetics are the observations that living things often posses traits that differ in seemily non-continuous ways (tall vs short, one vs three, yellow vs green etc).  Interestingly, it is possible to situate particulate Mendelian inheritance with discontinuous variability within morphometry and thus offer a new mathematical organon to instruct developments in population genetics.  Thom’s mathematical idea of structural stability can be used to conceptualize a specific construction of when genomic and phenotypic discontinuous variants possess causally the character of stability and thus function throughout changing gene pool representations of genes obtained generationally.  Stability is a particular property of Thom’s space of external parameters for pairs of dominants and recessives collected by categories of genome and phenome morphometric (macron) phenomenalizations.


 So what catastrophic metamodeling does for genetics is to show there is not a simple dualism between Galton’s view and Bateson’s interpretation of Mendelian division.  There is a catastrophic relation between reciprocal heterozygotes (retained embyrologically with complete genome doublings during histogeny) and their double homozygotic counterparts.  Dobshansky too easily reasoned from the existence of observational sterility and inviability  mutants  to that mutants arise from heterozygotes rather than homozygogtes regardless of future adaptability . It is the macrons which depend on the physical forces involved in complex combinations of subsistance attractors and repellors per basins and not the statistical division of categorical pheonmenalizations post hoc that realize the difference. The way genes and gene parts are identified with particular DNA lineages is a type of order categorized with Mendelian morphometry not total effective population size of any format. The Biometrican –Mendelian debate divide was simply due to the unknown genetic code behind the factors inherited and the lack of a creation of a mathematical horizon of bifurcation theory begun in large by Poincare to evolutionary theory. Fisher’s recognition of continuously varying traits explained by Mendelian inheritance is extended to parts of genes and now the outside environment is catastrophically mathematically localizable within the cell itself, thus allowing an expansion in the explanation of atomic differences (statistical mechanically) heritably. A general case of organism environment co-evolutionary dynamics is diagrammable with this  application of Thom’s morphogenesis to extinctions and adaptations when dominants reach ascertainable population levels no matter the kind or variety of point mutations that occur nor the kind of dominance evolved.

"The cosmic process uses unrestricted multiplication [14] as the means whereby hundreds compete for the place and nourishment adequate for one; it employs frost and drought to cut off the weak and unfortunate; to survive, there is need not only of strength, but of flexibility and of good fortune.

The gardener, on the other hand, restricts multiplication; provides that each plant shall have sufficient space and nourishment; protects from frost and drought; and, in every other way, attempts to modify the conditions, in such a manner as to bring about the survival of those forms which most nearly approach the standard of the useful or the beautiful, which he has in his mind."

Do blinking fractals (more/infinite multiplication than another direction) applied to classes of convergent  Galton-like series  model this difference between cosmology and horticulture narrated by Huxley (Evolution and Ethics 1894)? Can a catastrophe theory representation of  a grossone set of infinite series permit both continuous and alternative inheritance constructions in the same theory?  Is it possible to describe a distribution of offspring heritage into the gametes of individuals? Can social infrastructure selection be reduced to gametic reprensetations of moving forces dependent on amino acid contents coded and contributed by ancient ancestors? Can  a reason for gamete size differences, asexual lineages and different genders be developed out of such an organon?

We are thus in a position to acquire what Pearson related in plasma once the catastrophe theory delineates the relation of those traits that are alternative and those that form from changes in simple intensity. I have within my cognizance at least the veriest outline of the phenomenon!!

This may be possible without specifying the immediate cause of relative dominance. Vicariance provides a means to realize the inheritable latents and patents in a(the) systematic subtraction.

This grossone alteration of the infinite sequence Galton used follows Yule's view somewhat (but not with A a as substitutes for mean values).

  • Let us sup[pose that we have two different infinite sums –
  • This is the source of the historical controversy between the Mendelians and Biometricians (Mendelians – you can’t even necessarily have them – Biometricians – We have them and populations split them.
  • We can proportion this difference either within a single generation or between generations.

It appears that the given view is that Pearson and Weldon etc via Galton thought that populations split normally in a continuum regardless of how traits become expressed in the individual living thing.  The Mendelians supposedly contrarily had the idea that the factors which develop into the living thing’s phenotype come at least in discontinuous pairs and should not be thought ipsofacto as of  recombine ation  in different species via a continuous change in the traits.

Groosone infinitesimal subtractions when viewed catastrophically permit a phsyiological mehcanism  to be proposed for what the Biometricians never tried to do and yet this can be done without detracting from the Mendelian advance.

Rene Thom’s catastrophe theory of morphogenesis failed to gain any serious following as an organon for instructing research in biology.

 It had been thought that it might apply widely (Zeeman) and in particular to extinction (Lewontin Triple Helix) but this has never been realized in a robust practical circuit between theory and experiment.  Here we derive the Galton’s idea of Regressed Selection on the changing perspectives to distributions relative to a certain critical point (representing the average) and describe a model for stabilizing selection therethrough.  Different critical sets of higher order singularites are then introduced to modify Galton’s law (clearly defining differences of change of the mean/average) from the standard deviation thus decoupling the regressed correlation and making a broad discussion of cause and correlation possible.  This is used to explain better Wright’s notion path analysis as a different distributions shown by different catastrophe sets format change contributed both within and between generations for a given population.  Notions of potential and kinetic energy are supervienentized onto these coordinations of Mendelian factors and a means to understand how different genes get on different chromosomes is proposed through minimization of the energy function.  Application to sex- linked genes is provided.

So imagine a fold on the point of stablizing selection that opens a butterfly cusped wherein two unbilic alternate and organize these differences on a infinity computer and there you have it - a view of vicariant speciation organized both within and across generations by sets of grossone enumerated divisions.

In an attempt to understand how Pre-Darwinian, Pre-Modern Synthesis, and Pre-Punctuated Equilibrium periods of discussion are to be related we look at the two-humped camel representation of the Metamorphosis 3 of Catastrophe Theory as a presentation of the Weldon-Pearson decomposition of non exactly normal curves.  This presentation will allow for Fisher’s supposed resolution of the difference to be further expanded  so as to address Gould’s thought that Kimura’s view changed the terriotory that evolutionary theory will be stretched onto but which Wright felt rather needed to remain as a network rather than the aggregate of selective levels of ontology Gould imagined.  This model of the population allows direct physical forces to be imagined in the place (empty space) that Mendelians and Biometricians differ in .  It is a rational advance supplying what Henery Huxley could not imagine – Darwin’s evolution and Materialism of forces (the supposed “absurdity” between materialism and spiritualism is logically separated). Through a grossone alteration of Galton’s Law (subtracting an infinitesimal in the offspring’s heritage summation) on is able to introduce an infinite number of intermediate links between the changes (an infinity supplied by the catastrophe theory demonstation of the difference in recessive and dominance) so Galton’s senetence does not apply in this empricially verifiable proposition. SETH 346 (“Yet the changes are not by insensensible gradations; there are many but not an infinite number of intermediate links; how is the law of continuity to be satisfied by series of changes in jerks?”(1884 editiion , p 360).  Cantor’s continuous motion in a discontinuous space is applied to a grossone Calculus and levels of organization are described before one enumereates levels of selection intuitionally.

Here we offer a Catastrophe Theoy Model for proportionment both for between and among generations using grossone sums of infinite sums that decouples the between generation standard deviations from it mean.  The two processes (regression and change of average contribution) are organized by a vicariant pattern that may be formed directly from a given generations distribution of infinitesimal differeneces from unity or through variance in the vicariant criticality functioning between generatiohs.  Application of optimization at these two scales (as ideal extremes) is applied to social  selection sensu Roughgarden.  Morphological demomstration of a hypothetical sequence is serially derived for trilobites. Social coalitional size relative to populations is shown to depend on both the number of infinitesimal differences within the distribution within a generation as well as number of vicariant infinites between generations.  This possibility of differential proportionment appears to be the reason Provine (1971) asserted that there were two incompatible mathematical processes at work in the  SD01xo1/.5+SD12x11/.5^2+sd23x21/.5^3+…

The steps between changes do not necessarily HAVE TO BE small as Galton insisted but the inelastic divergences always occur where the compression is narrowing the space the penetration expands.  In that limit the discontinuity is as the Mendelians narrated but did not rationally manage to perfect to its purity as detailed here. Galton’s facet flipping can be intricate intended herein.  We do not need a different style of causality as Gould supposed however with this as the pool cue pusher.  Provine's tried to find B=C when A=B-C did not make sense.  B-C provides

Castle cases which do apply. Each (B-C) has a infinitesimal associated and an infinty on the inverse. A grossone interpretation helps to understand Galton's qualitative arguments.


Here we are able to show how Galton’s law remolded in a grossone system of modeling demonstrates cases where this change may be small or not so (difference of (1)+1, (1)+2…vs (1)+1,2(1), (1)^2).  Changes of the large size can still occur even when the pool cue pushes the fact through Galton’s law -1/2^{(1)} in this Mendelian discontinuity.  The repulsions and attractions in the coded amino acids etc will be worked in this narration about the mutation rates effective in the change where Galton narrated sport vs variation.

Thus this accomplishes what Bateson called for "directly attempt to giver any account of the distribution of the heritage among the gametes of any one individual."

The gamete and the germ-cell are different. The butterfly catastrophe operates between them before formation of the zygote and the combination of umbilic catastrophes.

Castle 1903 “In the year 1889, give precise mathematical expression to the well-known fact that the child resembles in varying degree its ancestors near and remote…He found that children resemble their parents, on the average, more closely than their grandparents, and the latter more closely than their great-grandparents, and so on to ancestors sill more remote.”

Thus if one admits to evolution than a change of forms would, on this math, exist  in limit of this process such that the next position after heritage a different form would exist.  This next is a post-heritage form and the relative position of the point after the limit need not be organized as to the order of  specific change.  In other words, Cantor’s finding that while the number of points in a linear continuum may be cardinally the same they can be ordinally different ( the rational numbers and the natural numbers have the same cardinality but different ordinalites).  Galton’s law on evolution suggests the investigation of finding different ordinal numbers for different for different limit sets within a particular sister divided monophyla.  It is strange that this viewpoint has never been pursued .  Perhaps it was due to Frege’s interpretation of Cantor. Nevertheless the finitistic view from this infinity Sergev enables practical ordinal differences that  hard to realize theoretically to be actually attached to different limits of Galton’s law through a superset of limited sequences organized around infinitesimal subtractions from the a unity a limit otherwise organized by Galton’s law as originally stated.


Castle wrote, “In all cases of alternative inheritance the person (or soma) represents only a part of the ripe germs produced by the individual, in some cases it may even represent none of them.  Hence any theory of heredity which bases its predictions as to the character of the offspring soley upon the character of the soma of the ancestors, is clearly inapplicable to cases of alternative inheritance.  The presumption is against its application to any other class of cases until that applicability has been demonstrated.”(p227)

The Grossone Modification of Galton’s Law under a catastrophe topology however can use the soma of the ancestors when the limit sets are correlated (need not be caused) with morphological form differences.  Thus with the trilobite example we can show the applicability of the infinitesimally different Galton law even while/if the heritability is cognized alternatively.  The theory of heredity that results is robust to this kind of criticism in general and may only fail if sex-linked trait caused evolution does not fit into the same bi-sexual split under vicariance.  Prima facie prejudice based on this argument of Castle and to some extent some of Bateson’s concerns are thus advanced beyond when Cantor’s continuous motion in a discontinuous space applies to the catastrophic topology that sorts the limit points reached immediately beyond somatically ascertained ancestral heritage of the fom-making.

The range of an alleomorph series will depend on how the butterfly catastrophe turns the relative streching distance of the umbilics (recessive and dominants) without needing to appear on a different chromosome.  Introns and exons can be expected in these places.Unless modifiers are inserted via the butteryfly catastrophe torque between the other two umbilic catastrophes it is unlikely that Fisher's view of the evolution of dominance is correct on this model.

How balancing selection is related to vicariance remains to be investigated.The hybrid heterozygote with paretnal in which all three catastrophes are present bears on the evolution of domaince through whether particular chemicals bear the force driven locus position or if it is purely in the difference in the energy levels bewtween potentially bonded atoms (electron vs place of electron) to Haldane's concept.
Direct visualization of catastrophe heeterozygotes may help to resolve this through higher order(dimensional) catastrophes. This may be constrained by the codeor else other kinds of genetic codes may be supposed where pre-DNA genetic factors were only in thought. It may possible to anticipate different codes that may be found on Mars.

This sythesis thus will resolve the of Galton's petty influences onto the Midparent and is subtracted by a infintesimal amount to resolve the difference of the
definition of the mid-parent and Darwin's notion of sexual selection Fisher made into a runaway. Thus regardles of the whether the screen off of the electrons or the places
of the electrons repositionable by chemical bonds (how D-seperation and latent variables affect results) the Grossone Sum allows one to differentiate social selection from sexual selection definintively and resolve why Rougharden
found Sexual Selection to be dogma being mined for confirmation rather than being tested against an alternative.  There is no possible alternative linguistically between the screening
off between sexes (what results in the potential of multiple genders rather than seperate sexes) and a topological movement outside of a regressed relationship.  This however is]
possible with a grossone calculus when Galton's law is reinserted into a Mendelian discontinuity via catastrophe sets.

This alternation of Galton's Law gives enough tolerance to the fit of offspring from predecessors towards the environment such that the habitat can be isolated enough wherein the ancestral energy in form-making continues into the present.  The separation of the distance of a single critical point into two combined in the dual hybrid contains the superficial surface on which active causes of form reside relative to absolute space.  Each complementary Mendelian (opposite) trait possess this surface without the actual cross bred hybrid as the body and the relative space moved OR the relative space at rest and the body moved. Recessivenenss and dominance are supported as the ridge lines that are incidence with the superficial surface but are penetrated genetically.  Kinetic and potential enegy contributions of the ancestors to the heitage of the offspring can be defined in that physiology of dual hybrid.  Higher order catastrophe sets make the construction of  more elaborate relations of dominace and recessivenbess to be combined.  This model seems to offer an atomically preferable structure to the pre-Darwinain thought that reasoned from metabolic inhibition or modifer genes for the same empricial data to be interpreted.  This new hybrid physiological process offers the possibility that biotechnology could grow different strains of living things with the same genes on different chromosomes and then put the multiplied individuals into a machine and extract the difference in the energy content possibly via the entropy relations in DNA computers.  Whether of measure of this difference could be designed into a machine to do work remains to be seen.  That was the goal to which this modeling of the moving forces of form-making retained across geneerations was directed.

Here I suggest that one might be able to rearrange each limited heritage by the number of folds cusp butterflied per distribution of rescessives and dominants in a gvien alleomorph series.  That challenge would be to compute fold cusp butterflied vicariances per heterzyogte and find the network that this travels for any given evolution of dominance (how the hybrid organizes the cusps fold per elliptic in a given population)

As this work is developed it will be possible to work out the relation of  "original form" to "force" and catalogue those moving forces that may be originally elastic or derived and when these are ordered by electro-magnetic force formats or merely through work done by impact forces no matter the pressure in the compression.  This will allow one to decide if the different forces in living things can ever be like Kevran suggested (utilizing the weak force in a special way),  offer evidence of Newton's reference to other kinds of forces and forms, and or merely be formats of distances known atom caused motions concuss (oscillation vs pulsation where genetics is a friable seperation Kantian wise).  The organization around Kant's ides of physics, math and philosophy in the Opus postumum can modify the following assessment.

Attractions and repulsions per density are aggregateable in to variable volume sizes dependent on the quanity and quality of the translations, reflections, and rotations amongst the distances involved. These create classes of left and right cetripetal and centrifugal sprials as the proportion of repulsions per attractions are distributed in an ordering of the different density  volumes observed in the given moving force.

Catastrophe Theory of the Geneotype – Phenotype relation.

While much has been made and lost about the hope and actuality of the application of catastrophe theory in biology, I would like to propose a fundamental role in its use to motivate concepts of the moving forces behind the geneotype – phenotype difference.  I use it to build an intuition about how ancestral energy is stored in current forms and dissect the formal forces differences (repulsion – attraction) as represented in DNA into arbitrary phenotype shapes through the  (7)elementary discontinuities and show how population genetic changes can be derived from attachment of different singularities to the  recessive – dominance trait scale.

Relying on a subsumtion of biometry into continuous darwian evolution historically viiels the need to go beyond Wright's simple idea ofd inactivation in recessives. It is not multiple factors per say that mechanically binds mendelism and biometric subtrations of Galton's law but rather the amount of attractions and repulsions coded into the recessive dominant pair per folds of any higher  order catastrophe sets giving the same volume as densities vary between the superficial and penetration surfaces. The philolosphy containing the propagation per organziation is much more detailed and determined than a simple emergence from a difference of a holism and a reductionsim.

The logic used on this page operates in a inverted way relative to FIsher's "reduction".  One first supposes a vicariance grossone modified and finds a division capable of operating the Mendelian dichotomization (oppositie motions) ( one with body at rest and a relative space in motion -- the other with relative space at rest and the body in motion for the same  absolute space ( purity, logic, math).  This has nothing to do with tasteful purpose.  Here biometry and mendelism are truely  equal both being presented as catastrophe points (mendelism utilizing umbilicis and biometgry the fold with the butterfly bridging where the multiple factor hypothesis resided.

The regression coefficient can occurr through the fold via the butterfly but the reversion can also appear from the umbilics.  This depends on the infiniteis associated with the infinitesimals that are subtracted (both the numnber of ininity summed to and the intensities summed). With vicariance assumed first rather than mendelism one is not limited to continuty and discontinuity of traits (dervied from a phonormic angle instead) only within and between "races" (geographic variants) as did Yule.

Individuals and gametes are very different ad Roughgarden in the Geneial Gene has well shown.  Gametic redistribution of ancestral purity rather than Yule's suggestion being prop for Fisher's theoretical  apparatus is possible.  External circustances can in some cases affect the conditions of A zygote presentation of TWO germ-cells.  It is possible to use the potential energy difference between a population of gametic DNA and zygotic  DNA.  This may occur facultatively through a friablilty effect of the conditions on the quantity of the constitutions doubled under physical seperation (in the extension wherein Biometricians were "constituionally unable").  Thus it represents a division forced on a shape persisting at rest.  Mitotic divisions and differentations are a particula instance of this.  It cannot be in motion unless the ahppening also depends on internal circumstances (genetic linkage etc.). Population genetics predicates nothing in regard to these gametes which go on to form bred zygotes, no matter whether the inhertiance is blended through a common scale or via relativity of motion in opppositions.  There are many behaviors associated with excessive gametic use activ ity and these heppenings have an evoltuionary origin not modeled in poulation genetics.   This is why Roughgarden had to suggest a reversed logic two tier theroreictical system.

So what are the possible constitutions of the gametes that exist in demes?

Asexual lineages, Multiple genders and homosexual behavior thus can be determined to directly attempt to give an account of the distribution of the heritage among the gametes of any one individual when a grossone mathesis is applied to the intensities of the ancestral law modified by a topological configuration of the texture of a recessive – dominance per double parent-hybrid heterozygote in a wild type.  The heterozygotic heritage thus depends both on the intensities and infinites summed and once suitable infinitesimal subtractions from the infinite intensity equation is made Mendel’s development binomial is approached precisely as Bateson surmised Aa with NaCl.  The differences in the ancestral contributions are due to different infinitesimal subtractions in line with dominance –recessive of the trait considered back of that path analysis.  The alternative A and a of the mendelian particularity is contained in the relation of these infinitesmials to larger and larger grossone constructions that do not house/shelter/contain the individual behavior, gender effective reproduction, or asexual lineage heritable but do contain space for such of further halving. Darwin’s version fails since this requires tetration directed rather than exponentiation checked but Bateson went too far to say that page 26 “It is difficult to see any reason for supposing that the manifestation of characters seen in the zygotes should give any indication as to their mode of allotment among the gametes.”

Continual halving and multiple gender formation across asexual lineages with homosexual behavioral equilinbria inbetween which operates nevertheless within Bateson’s “appreciation of the relation of zygotic to gametic characters” does grossone operate within constantly differentiable allotments among the gametes based on the ancestral heritage of all the zygotes path analysis back  of the current grossone sum. Transmission is possible in this frame.  Gametic purity and ancestral purity are different derivable from those numbers in a grossone system and the set of numbers derived from such a system.  True breeding can result from different combinations of gametic and ancestral purity which results in different gametic redistributions that cannot be reached from zygotic recombinations of the same elements since the infinite and infinintesimal are not reciprocally coordinated.  Multipe genders halving halved the binary supposed no matter how formed (small behavior large lineage) are organized division of the friable penetration of the variation and are uniform even though not homogenous in these divided sense that recessive vs dominant pure breeding is.  There is “displacement” of an alleomorph by another in this sense that within the DNA code the relative place may be replaced. HereDNA  computers could interface to extract the energy between the gametic structure and zygotic strucdture and mole bio.

The probable constitutions are thought to be determined by population genetics (loci location and frequency) but the versimultude of these differences can be governed by another constitutional possibility wherein the gametic distribution does not match the zygote combination.  Here the a halving of the population’s difference between homozygotes (recessive- dominants) can lead to multiple genders in the same linkage system which causes forces to be expressively compressed on the zygotic purity of reproductive effort and if the ancestral contribution is distributed to match the combination of the gametes leads to constraint towards asexual lineage formation when homosexual behavior externally is in sync.  Other relations of combinations and distributions are possible.

With this halving and multiple gender formation it is no longer assumable the recombinations that gave rise to the current pool of gametes in the heritages in any individual remain capable of transmitting ALL of the characters to the zygote even though population genetic probability indicates so and it is understood to follow by Galton or Pearson’s law.  This is because the intensity within and the  infinite demonstration of ancestral series are two different numbers that can sum to different values depending both on the intensity and how infinite the ancestral generations are between forms representing total population genetic variability.  Thus technically it is only the zygotes and not the gametes that can be said to possess a “fair sample of all the racial characters in their appropriate intensities.”  Eugenics was misplaced and mistaken because it did not realize this difference.

Thus depending on how sex (producing( quantitively in the quality)small or large gametes) is chromosomally dependent multiple genders endgendered behaviorally do result in gametic redistributions that split the difference of the heritage independent of the intensity per say even within an alternatively opposed inheritance.  There is no “transmission” thus in these cases but a redistribution that structurally changes the linkages not loci themselves.  When these previous behaviors are not lethal new genders can form and asexual lineages can be channeled.

[Forthcoming in The Journal of the History of Biology]


The “Evolutionary Synthesis” of George Udny Yule


James G. Tabery

Department of History and Philosophy of Science

University of Pittsburgh

Pittsburgh, PA 15260, U.S.A.

.Pearson, however, argued his own, “memoir of 1898 [discussing the law of ancestral heredity] adopted the simpler hypothesis that the correlation coefficients decrease in geometrical progression, it did not involve the fixity of the numerical constants of heredity which Mr. Yule tells us has not stood the test of time.  This simpler hypothesis…still seems to me to stand the test of time.”[i]  Pearson was claiming that he agreed with Yule in so far as there would be no strict application of universal constants to the ancestral law, but he disagreed with Yule in concluding that this meant there was no geometrical progression at all to the law. 

[i] Pearson, 1903, 229"


The grossone representation of ancestral purity is able to discuss differenece in the geometrical  progressive decreases of the initensities in the series where there are differenet infinite number of intensities with coefficients summed.  Thus this difference of Yule and Pearson can be bridged.  Priovine simply placed his own personal animosity in place of proper synthesis.  The analysis of the difference of math and philosophy in the transition to physics that Kant metphysically founded was missing from that historical retroperspective and  Will Provine conflated history and current thought. Pure gamete theory can deb desgined with grossone math applied across generations.  We are no longer thrown back while revisting Galton's law to the Biometrician of 1885 as Pearson analyzied with a better empirical approximation of the coefficients but rather rest satisfied that the history of 1985 was mistaken.  And for me personaly Will Proven's lack of action worked against my earlier advancment to this reather intricate dovetailling of the period my Grandfather learend evolutiuonary theory (prior to his PHD)in and passed on the thought to me from.

This misunderstanding on the biomath of ancestral vs gametic purity can be illustrated more clearly thorugh Sergeyev’s understanding of calculation of the area of the Serpinski carpet.  With a grossone system one can differentiate where one starts to compute the infinite steps in the sequence.  The difference of ancestral purity and gametic purity lies precisely in these different starting places. Interstingly Rouhgarden’s theoretical modeling cuts essentially orthogonally through this difference of opinion.

For example, instead of starting with a square, imagine starting with the shape in the top left had corner of the figure above, let’s call it a square doughnut. The square donut consists of eight squares, each with sides of length 1/3. Obviously, the area of this Sierpinksi carpet tends to zero as n tends to infinity.

But the square doughnut carpet is one step ahead of the traditional Sierpinski carpet but that gets lost in the traditional approach. At infinity they are treated as equal.

If that doesn’t sound very significant, imagine running the process in reverse, starting from infinity and working backwards to end up with a square or a square doughnut or some other shape in the carpet sequence. 

In that case, each shape can be created by the same (infinite) number of steps so it’s not possible to distinguish between them. That’s clearly absurd. 


Today, Yaroslav Sergeyev, a mathematician at the University of Calabria in Italy solves this problem (and the analogous three dimensional version called Menger’s sponge). 

For the last few years, Sergeyev has been championing a new type of mathematics called infinity computing. The basic idea is to replace the notion of infinity with a new number that Sergeyev calls grossone, which he writes like this: 

'Infinity Computer' Calculates Area Of Sierpinski Carpet Exactly

Mathematicians have never been comfortable handling infinities, such as those that crop up in the area of a Sierpinski carpet. But an entirely new type of mathematics looks set to by-pass the problem

Social infrastructure selection already did the by-pass. 

Both ancestral purity and gametic purity can exist.  Ernst Mayr was also mistaken when he told me that it would be typological and essentialistic to think of using the infinite differences within the serial representations of irrational numbers to catalogue the various minor variations between specimens collected.

Morgan's idea that both recessive and dominant aspects are within the hybrid and that parental forms do not appear is that aspect of the surfaces (not the ridges or singularites and are encoded in proteins by the overlap of the genetic  code between amino acids. Small ancestral mongrel purities can reappear in later generations through the grossone infintesimal seperation where small surfaces differences meet discontinuous amino acid changes that do not overlap and the attractive and repuslive forces are not compressed beyond the narrow limit the current monoplya purports rectinliearly.  It may still be possible to derive dominants that arise from recessives under this more intricate theorectical structure that was not available to Castle.  The inviability of the two yellow germ cells should be a case of the difference in energy between the gametic DNA and the zygotic DNA (compression overpowring expansion (repuslion and attraction in amiino acids) per penetration(lines layers and blocks gained despite exponential differentatioin)).

Castle made genetic facts trump the purely logical options possible.  In effect he played a game with the experiment and theory construction circle which confounded methodology and stoichiology.  He criticised Yule for claims that both Mendelism and Biometery could be developed concurrently and simultaneously made it a matter of the absolute acceptance whether alternative or continuous inheritance was real.  This thinking comes from a faulty relation of logic as a cathartic or as form thought and in truth is the only reason that creationism continued,  despite it’s attempt to get scientific, continued to be a major popular affector all the way to politics into the 21st century.  It fouled up the possibilities to untwist the relations of nature and nurture and prevented the synthesis of co-construction of the organism and environment.  In the end the differences of opinion on the evolution of dominance submerged the resolution and biology with its difference from physics has become its own worst enemy. The possibilities of small errors and complementarity can have the same topological ontology and it can be epistemologically accessed if we follow with Cornell’s idea of new technique for a body of functional hubs that interact the built environment, within connective media that leads to a healthier life.  The base of this is a new kind of purposive functionality in the soma that further explores the relation of the geneotype and phenotype and enables us to see that what we are working on today will be what we will hand down to our posterity in 10 years or 100 years from now.  We will know what technology will build and we will know it even better as it becomes a part of our body.  The internal phenotype is simply not recognized as outside the  environment yet within the occurring genetic motion that is prior to the moving forces that are engineer able by any hub.

Understanding the phenotype as a common surface of contact in Kant's metaphysical sense is rather fruitful.Heribitble Infinitesimal repulsive differences when grossone reciprocated establish infinite evolvabilities within the distances streched by the start and stop codons of genes.  The organism is not a passive vehicle as Dawkins had it in the 70s but rather is the motion from which the genetic revolutions force changes ofhappen dynamically in.


Consider the multiplicative property o f life to presented as an expanding molecular substance of one of four kinds (A,U,G,C).


The immediate effect of one kind on another is produced by the duplication of any one creating a common boundary of repulsive and attractive forces. The mutual  approach of any four always ocurrs in the inverse proportion to the quantity of the expansion (ends of the start and stop messaging as the gene) but this is purely mechanical.


When expansions of all four combined result one is able to garner the proportion of the attractive forces themselves.  This can be associated with Darwin's notion of multiplicative increase per lineage since the attractions are all in a line with the repulsions orthogonal but now this is not visible in the chromosomal format but only in the phenotype of the bounded moving forces of all molecular interactions within the cells. Cellular cohesion thus contains both the derivative and original elasticity the genes penetrate and can only compress the expansion so far as the starts of every mechanical directions are eliminate.  The fixation of genes is thus penetrative and orthogonal to the species' line of descent.


The attraction is the source of the penetration and is primarily genetic while the repulsion not less fundamental is rather only a superficial force but is the principle restraint on the expansion and is mostly phenoytpic.  It affects the proportion of the evolvable attractions. What is difficult to know is how this attraction in its total force field is or is not independent of the general gravitational attraction as this bears on how living things may interact with other forms of potential life from other planets under different gravitational regimes and different codes for the expanded substances across the mutually related distance boundaries no matter the differences in the volume to density ratios.

Evolution is a consequence of substances that both are expandable and can be diminished as they move from their places.  Evolution is life’s occupation of space.

Life is a substance capable of resisting every other matter that endeavours by its motion to press into its expandable space.

If a substance extends itself it does so by keeping others from its place.

In order to duplicate itself a substance either attracts a copy of itself into its adjacent space or repels others from its place wherein  only others of its own kind can thus reside.

If a substance has already approached itself it is in a double space.

When two lineages cross, attractions retained in the course of expansion are put against the repulsions established onto the environment but because the genetic code is not homogenous (some amino acids coded by 1,2,3, or 4 code schemes), these cross path trajectories can yield kinematics of concussive moving forces that likewise affect biotic and abiotic material depending  on  the action at a distance means signed by the code.

There are only 3 different direct lines whereby a substance once doubled can cause the approach of others to itself or resist the approach of others to itself.  If these lines are populated by resistances they fill the space of self multiplication but if by attractions instead they only enclose the space.  The code results in specifications of fillings vs enclosures.

Evolutionary theory is a means to indicate what effect the occupation of space has on both its abiotic and biotic neighbors. Resistance towards others not specific to it and/or creation of places without abiotic impenetrability filling the neighborhood  ensue.

Genes present different elasticities in so far as the starts and stops expand beyond what is the default in the code ( if stops of one of the two others not used per message where to be used at each doubling step). The 3:1 Mendelian dominant to recessive is based on the the three lines and 3 stops per 1 start and the dimensionality of the geometric line itself.  There is never a smallest or largest degree of extension in the heterozygote.

Cellular differentiation with its selective gene expression compress the space otherwise not only penetrated but expandable to, as if all genes were certitus paribus expressed. It presents a greater moving force than that molecularly physical within that of the code itself.

The apparent conflict between evolution of life from a common origin and separation into lineages is simply a matter of those substances’ moving forces having been able to not disallow further penetration of already occupied spaces compounded phoronomically from removal of abiotic lines by other moveables whether in the opposite direction or not.

If these three lines are represented by the shape of some other object it may be demonstrated by a triple repulsion-attraction.

A single point neutral mutation creates a “torque” on the symmetrical reflection (difference in translation and rotation).  Multiple chromosomes ensure a single inertial line but nevertheless a limit to the interaction of the rotation and translation.  Thus neutral phenotypic substitutions provide a basis for moving force changes by providing internal motion from which moving forces act during the motion of the same from its spaced place. There is no mechanical need for selfish DNA gene vehicle ideas as that was a notion due to lack of inertial application when gravity as a moving force is voidable/movable to a different place.

Because of this capacity living traits are able to resist the filling by anything but its own life in two oppositie phoronomically determined kinematically mechanical directions.  This gives rise to alternative inheritance as life propagates and expands itself which is quantitative through genes that so start one way but stop 3 ways in enclosing the fillable space.  The living boundary is only enclosed by the phenotype but it is filled by the geneotype.  This is the occupied space of life’s place.

Gene start-stop messaging as means to untwist phenotypic oscillations/concussions through divisions of attractions per ½ repulsions via the belt trick.

Repeat induced point mutations may be due to repeats finding locations where belt tricks no longer trick and thus increase the attraction to repulsions ratio.

The anchor end of the belt trick is where the methylation as a start is or where the meyltation outside the double attraction strand aggregates.

The need for both RNA and DNA may be in the friable separation of ½ the repulsion per attraction to prevent perverted torques.  Viruses attempt to propagate with only attraction and repulsion (not divided) and hence suffer twists which limits expansions otherwise compressible.

Is not the thought that G-C content being addabloe in a non-commaless code cocept differentially perhaps be a fact consequent to the intrication of differen ces attractions force impact through a finite difference equation (no matter the gravity) of G-C hydrogen bond distance vs. A-T(U) bonding (reflexion of ½ repulsion).

Without the wobble rules and two repulsions parts divided by RNA (Messenger and Transfer) the placers would only be enclosable and not fillable.  This would not submit to a universal response to gravity which is possesed by the difference in the code between the body general and the mitochondria which are subject to the gravitational force.  That is the volumes and the general density correlation is constrained by both the amino acid forces as with the DNA and RNA codes (including the tRNA stereochemistry.  There is no "carrying capacity' for this expandability but only a relation to gravity as related to metabolism genetics which can move in directions away from inertial effects.  If microbes are found on Mars they may posses a different relationship between the general code and a gravity affected code sequestered differently than on Earth.

Nim addition with its binary exclusive either or may be responsible for the 3:1 Mendel number and 3:1 stop to start message reading codons while nim multiplication applies to seperate not only the attractions and repulsions (binary) but the attractions and 1/2 repulsions ordinally per ordertype of  penetration.

Sex may actaully be propated chromosomally because of the difference between gravity and e-m as the moving force between nim addition and multiplication represented.