REVIEW 4 major objections 3 minor 3 references
Cosmology of Plane Geometry
T0 review · 4 major / 3 minor · reviewed 2026-08-14 · deepseek-v4-flash
Pith's one-line read The deformation principle says coincident objects in a special figure predict how their deformed versions connect in the general case.
desk verdict A likeable but unrefereable sketch: the deformation principle is an underdetermined heuristic, and every nontrivial claim is asserted without proof. read the letter →
The pith
A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.
The reading
What carries the argument
The machinery is the deformation principle itself: take a configuration in which certain objects coincide, then replace each coincident object by the same kind of object built by the same construction but no longer coincident. The principle says the degeneracy is informative: it encodes which points, lines, or circles will be connected in the general case. The paper's table of undeformed-versus-deformed pairs makes the mechanism explicit: coincident points may become collinear or concyclic points, coincident lines may become concurrent lines, coincident circles may become concurrent or coaxial circles, and coincident triangles may become perspective triangles or triangles on a common conic.
What would settle it
Test Theorem 1 numerically: choose a random convex quadrilateral with exact coordinates, construct the four isosceles-right-triangle vertices, and check whether $O_{ab}O_{cd}\perp O_{bc}O_{da}$ and $|O_{ab}O_{cd}|=|O_{bc}O_{da}|$; one failure would refute the deformation principle as stated. The same check applies to Example 1, where the centers of the outward equilateral triangles and the first Fermat point should be concyclic for every scalene triangle.
Extended reading notes
Core claim
The paper's central claim is that the deformation principle is a reliable engine for plane-geometry results: whenever an undeformed configuration has points, lines, or circles that coincide, the corresponding deformed configuration should have those objects connected by a named relation such as collinearity, concyclicity, concurrence, perpendicularity, or equality of lengths. The square with its center, seen as four coincident isosceles-right-triangle vertices, deforms into Theorem 1: in any convex quadrilateral these four constructed points satisfy $O_{ab}O_{cd}\perp O_{bc}O_{da}$ and $|O_{ab}O_{cd}|=|O_{bc}O_{da}|$. The equilateral triangle with its center, incircle, and circumcircle deforms into three asserted examples: equilateral triangles on the sides have centers forming an equilateral triangle through the first Fermat point; the second Fermat point of the base triangle lies on the circumcircle of the second Fermat points of three subtriangles; and reflected nine-point centers, together with the nine-point center, lie on circles. These statements are presented as direct consequences of reading the degenerate case.
Load-bearing premise
The load-bearing premise is that a coincidence in a special configuration must correspond to a single, natural relation among the deformed objects in every generic configuration; the paper does not prove that such a relation always exists or that it is unique.
Editorial extensions
If this is right
- Theorem 1 follows: for every convex quadrilateral $ABCD$, the isosceles-right-triangle vertices $O_{ab},O_{bc},O_{cd},O_{da}$ satisfy $O_{ab}O_{cd}\perp O_{bc}O_{da}$ and equality of the two lengths.
- For every triangle, the centers of the three equilateral triangles constructed outward on its sides form an equilateral triangle whose circumcircle contains the first Fermat point.
- For every triangle, the second Fermat point $F_2$ lies on the circumcircle of the three second Fermat points of $F_1BC$, $F_1AC$, and $F_1AB$.
- For every triangle and point $P$, the reflections of the nine-point centers of $A'BC$, $B'CA$, and $C'AB$ across the sides are concyclic with the nine-point center of $ABC$, and the same circle claim holds for the reflections across the side midpoints.
Reading between the lines
- A formalized version of the deformation principle could be expressed as a continuity or limit statement: if a relation holds on a dense set of degenerate configurations, it should persist under deformation; proving such a statement would convert the heuristic into a theorem generator.
- The principle suggests a practical discovery procedure: start from any known configuration with coincident objects, deform it symbolically, and test the predicted relation by computation; the three examples are only the first outputs of such a search.
- The scope may extend beyond points, lines, and circles: the table in Section 2.2 already lists conics, and the same logic could apply in projective or inversive settings where coincident objects force incidence relations under deformation.
Signed reviews
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
Summary. The paper proposes a 'deformation principle' for plane geometry: when a special (degenerate) configuration has coincident points, lines, or circles, one can 'predict' relations such as collinearity, concyclicity, or concurrency among the corresponding objects in a general configuration. The manuscript illustrates this with a theorem about four isosceles right triangles built on the sides of a quadrilateral, and with three numbered examples concerning equilateral triangles built on the sides of an arbitrary triangle, the Fermat points of a triangle, and nine-point centers of a triangle and its circumcevian triangle. The final section states that the full version of the article is available only through external links (Scribd documents), and the present note is explicitly a first chapter. The paper contains no derivations or proofs for the asserted geometric facts; each example is introduced with 'And, in fact' followed by the claimed relation.
Significance. If the deformation principle were made precise and its predictions were backed by proofs, this could offer a heuristic perspective that connects degenerate configurations to general theorems in plane geometry, and the specific examples (especially the Fermat-point and nine-point-center relations) are concrete statements that could be of interest to geometers. The paper's strengths are its explicit statement of a method and the concreteness of its examples, including the stated Theorem 1. However, the manuscript as submitted does not provide any verifiable derivation: the central principle is not defined with enough precision to be tested, and the examples are asserted rather than proved, with the actual content relegated to non-archival external documents. The paper also does not provide machine-checkable proofs or reproducible code. Thus, while the examples may be true (and some are standard results), the scientific contribution of this short note cannot be assessed from the text alone.
major comments (4)
- [Section 2.2, Table 1] The deformation principle is underdetermined. The table lists multiple admissible deformations for the same type of degenerate object: coincident points may become collinear, concyclic, or lie on the same conic; coincident circles may become concurrent, coaxial, or share a radical line. No uniqueness, minimality, or selection criterion is stated. Therefore, from a degenerate configuration alone one cannot predict which relation will hold in the general case. This is load-bearing because every example in the paper is presented as a direct consequence of reading off the degenerate case.
- [Section 2.3, Examples 1-3] The examples assert geometric facts without proof. For instance, Example 1 claims that the centers Oa, Ob, Oc form an equilateral triangle whose circumcircle passes through the first Fermat point; Example 2 claims that F2 lies on the circumcircle of FaFbFc; Example 3 claims two distinct concyclicity statements. In each case the paper only says 'And, in fact' and states the result. No derivations, references to proofs, or verifiable computations are supplied. Since the paper's central claim is that these facts follow from the deformation principle, the absence of proof is a major derivation gap.
- [Section 2.3, Example 3] Example 3 illustrates the underdetermination most sharply. In the degenerate case, the paper states that N'a = N'b = N'c = N = P = O and similarly N''a = N''b = N''c = N = P = O, so all six points in each family coincide. This single degenerate input is used to predict two distinct concyclicity statements: that N'a, N'b, N'c, N lie on the same circle, and that N''a, N''b, N''c, N lie on the same circle. The degenerate case gives no reason to privilege exactly these two circle relations over, say, collinearity of the same points, or a single circle containing all five points, or any other relation consistent with total coincidence. The principle thus cannot be checked from the text, and the asserted relations require independent geometric proofs that are not provided.
- [Section 3] The full content of the paper is explicitly relocated to external Scribd documents. A journal manuscript must be self-contained enough for the reviewer to verify the central claims. Since the present note states that the complete article is external, and the external documents are not part of the arXiv submission, the claims in Examples 1-3 and Theorem 1 cannot be checked from the submitted material. This is a structural issue: even if the deformation principle were made precise, the paper as written delegates its entire substance to non-archival, non-reviewed sources.
minor comments (3)
- [Throughout] There are numerous typos and formatting issues, including 'configuration' for 'configuration', 'Aditional' for 'Additional', inconsistent use of 'wrt' for 'with respect to', and extra spaces in phrases like 'circumcircle ( ABC )'.
- [Section 2.3, Example 2] The statement 'the next such relation can be formulated: Point F2 lies on the circumcircle of FaFbFc' is not preceded by a formal statement number or a proof; labeling it as an example rather than a theorem makes the lack of justification more conspicuous.
- [References] The reference list contains only three items, with two of them being the author's own earlier preprints and the Kimberling ETC link. The external full-version documents are cited only via Scribd URLs, which are not stable or peer-reviewed sources.
Circularity Check
No circular derivation chain: the deformation principle is an underdetermined heuristic, and the geometric examples are asserted external facts rather than reductions to fitted inputs or self-citations.
full rationale
The paper's load-bearing principle (Section 2) is the deformation heuristic: "If one is aware that points, lines, or circles are equal in the case of non-deformed configuration, one can predict the deformed versions' connections in terms of general configuration." This is not a definitional equivalence or a fitted input; it is an informal rule for generating conjectures. The examples (Theorem 1 and Examples 1-3) are stated as "in fact" geometric facts, but no derivation from the principle is supplied, and the principle is underdetermined (Section 2.2 lists multiple possible deformations of coincident points). That is a problem of evidentiary support, not circularity: the claimed results are not equivalent to their inputs by construction. No parameter is fitted to a subset and then renamed as a prediction. The references to Kimberling's ETC and the author's earlier arXiv preprints [2,3] are not load-bearing: [2] and [3] appear only in the reference list, not in any argument, and [1] is an external catalog. Section 3 points to a longer external article for proofs; the absence of proofs in this note is a completeness gap, not a circular step. I therefore find no significant circularity.
Assumptions & free parameters
assumptions (2)
- ad hoc to paper Deformation principle: coincidences in a special configuration imply specific relations (collinearity, concyclicity, concurrency) in any deformed configuration.
- domain assumption The geometric statements in Examples 1-3 are true.
Cite this review
Pith. "Pith review of Cosmology of Plane Geometry." pith.science (2026). https://pith.science/paper/DHJZY7EZ
@misc{pith2026190805084,
author = {Pith},
title = {Pith review of: Cosmology of Plane Geometry},
year = {2026},
howpublished = {\url{https://pith.science/paper/DHJZY7EZ}},
note = {Machine review of arXiv:1908.05084}
}
read the original abstract
This paper focuses on a new approach to plane geometry and develops important concepts that can allow researchers to unite and observe plane geometry from a new, meaningful perspective.
Reference graph
Works this paper leans on
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[1]
C. Kimberling , Encyclopedia of Triangle Centers – ETC , http://faculty.evansville.edu/ck6/encyclopedia/ETC.html
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[2]
Some new theorems in plane geometry
A. Skutin , Some new theorems in plane geometry , arXiv preprint arXiv:1704.04923 (2017)
work page Pith review arXiv 2017
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[3]
A. Skutin , Creative geometry, arXiv preprint arXiv:1802.03543 (2018)
work page Pith review arXiv 2018
Reviewed August 14, 2026 · model on record in the stance chip above.
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