Elements of finite geometry I

Oliver Knill

REVIEW 2 major objections 5 minor 202 references

Elements of finite geometry I

T0 review · 2 major / 5 minor · reviewed 2026-08-10 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗

Pith's one-line read The finite theory reproduces continuum geometry: index-expectation curvature on fine triangulations agrees with the Gauss-Bonnet-Chern integrand.

desk verdict A readable, honestly footnoted digest of Knill's prior discrete-geometry results; the discrete math mostly checks out, but the claimed continuum bridge in §3.19 is asserted, not derived. read the letter →

arxiv 2608.06405 v1 pith:G2B7G45O submitted 2026-08-03 math.HO cs.DMmath.CO

Oliver Knill This is my paper · ORCID

classification math.HOcs.DMmath.CO MSC 05E4552B7055U10

keywords finitegeometrysimplicialcomplexesgraphsGauss-BonnetPoincaré-HopfindexexpectationEulercharacteristicWu

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

Finite geometry—graphs, finite abstract simplicial complexes, and delta sets—carry theorems like Gauss-Bonnet and Poincaré-Hopf with no limits or regularity assumptions. The paper's twelve units present exact finite versions of Gauss-Bonnet, Poincaré-Hopf, Euler-Poincaré, Brouwer-Lefschetz, sphere formulas, level-set behavior, and higher characteristics, all tied together by index expectation: the average of Poincaré-Hopf indices over random functions defines a curvature that sums to Euler characteristic. The strongest claim is that the finite theory is not merely analogous to the continuum: for a sufficiently fine triangulation of a compact Riemannian manifold, the index-expectation curvature built from linear functions on the ambient Euclidean space becomes the Gauss-Bonnet-Chern integrand. If true, the discrete theorems genuinely reproduce continuum geometry.

What carries the argument

The central mechanism is index expectation, $K(v) = E[i_g(v)]$, where $g$ ranges over random locally injective functions (colorings) on the vertices and $i_g(v) = 1 - \chi(S_g^-(v))$ is the Poincaré-Hopf index; here $S_g^-(v)$ is the subgraph of the unit sphere of $v$ consisting of neighbors $w$ with $g(w) < g(v)$, and the unit sphere is the graph induced by the neighbors of $v$. Linearity of expectation converts the pointwise Poincaré-Hopf identity into a Gauss-Bonnet identity, so curvature becomes a probability-space object. In the continuum limit, the random functions are taken to be linear height functions from an isometric embedding of the manifold into Euclidean space, and the same expectation is argued to equal the unique local invariant curvature form of the continuum.

What would settle it

Take an explicit fine triangulation of a compact even-dimensional Riemannian manifold, for example the 4-dimensional ellipsoid in $\mathbb{R}^5$ displayed in Unit 3, compute the index-expectation curvature $K(v) = E[i_g(v)]$ using linear height functions $g(x) = x \cdot a$ with a uniform distribution on directions, and compare pointwise or integrated against the Gauss-Bonnet-Chern integrand; if the difference does not tend to zero as the triangulation is refined, the central claim is false.

Watch

Extended reading notes

Core claim

The paper claims that Euler characteristic, curvature, index, cohomology, and fixed-point formulas usually stated for smooth manifolds hold in the same form on finite graphs and finite abstract simplicial complexes, by purely combinatorial definitions involving no limits. The load-bearing mechanism is index expectation: choosing a random locally injective function and averaging its Poincaré-Hopf indices produces a curvature $K(v) = E[i_g(v)]$ that sums to $\chi(G)$. When the finite complex is a sufficiently fine triangulation of an isometrically embedded compact Riemannian manifold and the random functions are almost all linear height functions from the ambient Euclidean space, this discrete curvature is locally homogeneous and therefore must be the Gauss-Bonnet-Chern integrand by a uniqueness argument for local invariant curvature forms. The paper presents twelve results using this mechanism: Gauss-Bonnet, Poincaré-Hopf, index expectation, Euler's gem, Euler-Poincaré, unimodular connection matrices, Brouwer-Lefschetz, sphere formula, level sets, index formula, quadratic cohomology, and higher Green functions.

Load-bearing premise

The load-bearing premise is the unproved bridge in §3.19: the discrete index-expectation curvature built from linear height functions on a fine triangulation of an isometrically embedded manifold is assumed to equal the Gauss-Bonnet-Chern integrand by a uniqueness argument, with no derivation supplied; if that bridge fails, the finite theorems remain internally true but no longer imply the continuum results.

Editorial extensions

If this is right

Finite graphs and simplicial complexes carry exact, limit-free versions of Gauss-Bonnet, Poincaré-Hopf, Euler-Poincaré, Brouwer-Lefschetz, and fixed-point theorems, with the same Euler characteristic as the continuum object.
If the discrete-continuum bridge holds, any even-dimensional compact Riemannian manifold can be approximated by finite graphs whose index-expectation curvature converges to the Gauss-Bonnet-Chern integrand, giving a combinatorial route to the Chern-Gauss-Bonnet theorem.
Odd-dimensional manifolds are flat in this theory: for any symmetric probability space invariant under $g \mapsto -g$, the index-expectation curvature is identically zero, and in 4 dimensions curvature is the expected genus of a random surface in the unit sphere.
Every function on a finite manifold has level sets that are manifolds (or empty) with no regularity condition, so the discrete Sard phenomenon is exact: singularities do not occur.
Higher characteristics (quadratic, cubic, k-particle) are barycentric-refinement invariants, hence topological invariants, but not homotopy invariants; for manifolds with boundary they reduce to Euler characteristic plus boundary correction terms, and their k-point Green functions sum to the characteristic.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

Editorial extension: if the bridge holds, a practical numerical scheme suggests itself: approximate Riemannian curvature integrals by counting local extrema of random linear functions on fine triangulations, without ever constructing Riemannian metrics or connection forms.
Editorial extension: the identification of 4-manifold curvature with expected genus of random surfaces invites a quantitative test, comparing index-expectation curvature to scalar curvature on a dense triangulation of a 4-manifold to see whether the random-genus interpretation matches known curvature functionals.
Editorial extension: the k-point Green functions defined on all k-tuples of simplices, finite and singularity-free, are natural combinatorial analogues of correlation functions; one could ask whether their large-complex limits reproduce familiar Green's functions, such as logarithmic or inverse-power potentials, in Euclidean spaces.
Editorial extension: the uniqueness of the barycentric-invariant valuation suggests a classification principle: if all higher characteristics are forced by symmetry alone, then any alternative finite geometry agreeing on complete complexes must be a multiple of the corresponding characteristic.

Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Short version: this is a clean, honestly footnoted digest of twelve discrete-geometry results, almost all from Knill's own earlier papers, and most of the discrete mathematics is transparent and correct. The one load-bearing gap is the discrete-to-continuum claim in §3.19, where the text asserts that index-expectation curvature converges to the Gauss-Bonnet-Chern integrand without giving the proof or a precise reference. The stress-test note gets that right.

What the paper actually does well: it gives short, checkable proofs of discrete Gauss-Bonnet, Poincaré-Hopf, Euler-Poincaré via supertraces, the Green-star formula, the sphere formula, and the level-set theorem. The energy-distribution arguments are clean. The worked examples (star graphs, Platonic spheres, K3 complex, dunce hat, Möbius strip) are concrete and reproducible, and the footnotes attribute each result to its source. As a digest, it is honest: the reader is told which unit comes from which preprint.

The soft spots are real but localized. Section 3.19 is the only place where the paper claims the finite theory reproduces continuum geometry. It invokes Nash embedding, ambient linear functions, matching indices, local homogeneity, and 'an argument of Weyl' to conclude the index-expectation measure must be the GBC integrand. Missing are the actual convergence statement, the locality argument for the expected index current, and the uniqueness theorem applied in the right setting. The footnote that discrete curvature 'produces' the GBC integrand repeats the claim instead of proving it. If the bridge fails, the discrete theorems remain valid internally: Gauss-Bonnet and Poincaré-Hopf still hold for every finite complex, and level sets are still manifolds without Sard assumptions. What weakens is the introduction's broader claim that finite geometry reproduces continuum geometry.

Two smaller issues: the Platonic-sphere classification in §4.26 is compressed to the point of being a sketch, especially the exclusion of the 600-cell as a unit-sphere type for a 4-sphere. And names like 'Hydrogen operator' and 'scattering amplitudes' are motivational analogies, not physics claims; harmless, but they should not be mistaken for results.

Who is this for? Someone who wants a single-entry overview of Knill's discrete-geometry program, or a teacher looking for short, elegant proofs of classical discrete theorems. It is not the place to find new theorems. It deserves a serious referee: the discrete core is mostly sound, and a referee could usefully demand that §3.19 be either proved, given a precise published reference, or explicitly labeled as a conjecture.

Referee Report

2 major / 5 minor

Summary. The manuscript is a survey/snapshot of twelve units in finite geometry, treating finite simple graphs and finite abstract simplicial complexes. It develops combinatorial Gauss-Bonnet, Poincaré-Hopf via index expectation, Euler's gem for q-spheres, Euler-Poincaré via Hodge/Dirac/McKean-Singer, unimodular connection matrices, Lefschetz fixed point theory, the sphere formula, level-set theorems, the symmetric index formula, quadratic cohomology, and higher Green functions. Many theorems are proved by short energy-distribution, generating-function, or heat-flow arguments. The introduction and Unit 3 additionally claim that the finite theory is not merely analogous to but actually reproduces continuum Riemannian geometry: index-expectation curvature on fine triangulations of a Nash-embedded compact manifold is said to equal the Gauss-Bonnet-Chern integrand. This continuum bridge is asserted rather than proved, and a second classification claim, the uniqueness of Platonic spheres for q>3, is also presented with only a sketch of a proof.

Significance. If the discrete theorems stand, the paper offers a coherent and largely elementary development of finite geometry with transparent proofs: Gauss-Bonnet by energy distribution, Poincaré-Hopf by divisor colorings, Euler-Poincaré by supertrace heat flow, and the explicit Green-star inverse L^{-1}=g are attractive and checkable. The unimodularity theorem, the level-set theorem, and the higher Green identities are distinctive contributions. However, the advertised significance of the project, that the finite theory 'produces' the continuum Gauss-Bonnet-Chern geometry, rests on an unproved convergence statement in §3.19. The paper's value as an expository survey of the discrete theory is real, but its central external claim needs either a proof, a precise reference, or an explicit demotion to conjecture.

major comments (2)

[Unit 3, §3.19] The load-bearing identification of index-expectation curvature with the Gauss-Bonnet-Chern integrand is asserted, not proved. The passage invokes a Nash embedding, ambient linear functions, local injectivity on sufficiently fine triangulations, matching Poincaré-Hopf indices, and 'an argument of Weyl' that the expectation 'has to be' the Gauss-Bonnet-Chern integrand. This does not establish the necessary analytic claims: that the expected index measure E[Σ_p i(p)δ_p] is a locally defined absolutely continuous density on M; that it is a Riemannian invariant independent of the triangulation and of the embedding; and that it converges weak-* to the GBC integrand under refinement. Weyl/Gilkey uniqueness of invariant differential forms does not by itself supply existence, locality, or convergence for the index-expectation construction. Without these steps, the introduction's 'discrete implies continuum' claim is unsupported. The manuscript should either supply these arguments or explicitly mark the statement as a conjecture.
[Unit 4, §4.26] The classification theorem 'There is a unique Platonic sphere for q>3' is not proved. The q=4 case asserts that the only possibility is the 4-dimensional cross polytope and rules out a unit 600-cell by a one-line computation K=1, followed by 'Gauss-Bonnet would give |V|=2 and dim(G)≤1'; neither the exclusion of other Platonic 3-spheres as unit spheres nor the induction from q=4 to all higher dimensions is derived. As stated, this is a classification claim with a sketch, not a theorem with a proof. It needs either a complete combinatorial proof or an explicit reference to the standard classification of regular polytopes, together with a clear statement of what remains conjectural in the recursive definition used here.

minor comments (5)

[Unit 7 title] The unit title contains a typo: 'Brower-Lefschetz' should be 'Brouwer-Lefschetz'.
[Unit 3 diagram, Continuum panel] In the continuum diagram, the displayed formula '∫_M K(v) dV = χ(G)' uses χ(G) for a continuum manifold M; the notation should be χ(M).
[Introduction] The sentence involving 'GP= 10 (10100)' appears to have a formatting error in the googolplex discussion; the notation should be repaired.
[Unit 10 figure] The figure states that 'Euler characteristic of a 4 manifold is the difference of the volume and a Hilbert-Einstein type action', but this statement is not explained or used in the text; it should be removed or clarified.
[Bibliography and §3.19] For the Weyl/Gilkey uniqueness invoked in §3.19 and for the regular polytope classification invoked in §4.26, specific page or theorem references should be given rather than relying on narrative references to the author's own preprints.

Circularity Check

0 steps flagged · score 2.0 of 10

No significant circularity: the discrete derivations are self-contained; the asserted discrete-to-continuum bridge in §3.19 is under-proved but not a circular reduction.

full rationale

The twelve units prove their main discrete statements from the stated definitions: Gauss-Bonnet and Poincaré-Hopf are double-counting identities; index expectation follows by linearity of expectation; Euler-Poincaré follows from the McKean-Singer supertrace; unimodularity, the sphere formula, level sets, the index formula, quadratic cohomology, and higher Green theorems are derived by valuation, join, or heat-flow arguments already in the text. The only load-bearing passage that is not derived is §3.19, where the claim that index-expectation curvature 'has to be the Gauss-Bonnet-Chern integrand' rests on an asserted local homogeneity and an external Weyl uniqueness argument; the Unit 1 footnote refers to the author's earlier index-expectation papers for the verification. This is a real rigor gap: no weak-* convergence or locality proof is exhibited, and the conclusion is not established in this paper. It is not, however, circular in the technical sense: no parameter is fitted from the target curvature, no defining equation is equivalent to the conclusion, and the uniqueness theorem is attributed to Weyl/Gilkey rather than imported from the author's own prior work. The heavy reliance on self-citations for the continuum bridge is the reason the score is not zero, but the central discrete theorems retain independent content.

Assumptions & free parameters 0 free parameters · 4 assumptions · 2 invented entities

The central claims rely on standard linear algebra and topology plus a small number of program-specific definitions. The main uncharged items are the continuum bridge (§3.19), the recursive sphere classification (§4), and the nonstandard level-set definition (§9). No fitted numerical parameters appear.

assumptions (4)

domain assumption Nash embedding theorem and the Weyl/Gilkey uniqueness of Gauss-Bonnet-Chern type invariants.
Invoked in Unit 3, §3.19 to equate index expectation with the GBC integrand; no proof or precise reference supplied.
domain assumption The recursive definition of contractibility gives a decidable, consistent notion of q-manifolds and q-spheres.
Unit 4, §4.22-4.26 and the Structures section base the Euler Gem and Platonic sphere classification on this recursion; the text asserts decidability but does not prove equivalence with other manifold notions.
standard math The heat-flow argument (McKean-Singer symmetry) is valid for finite Hodge Laplacians.
Used in Units 5 and 7 to derive Euler-Poincare and Lefschetz; standard linear algebra but not proven in the text.
ad hoc to paper The level-set theorem's definition of M_g via the Whitney complex of the graph of intersecting simplices is the intended 'manifold' notion.
Unit 9, §9.59-9.60; the claim that all level sets are manifolds depends on this nonstandard definition, which differs from ordinary vertex-level sets.

invented entities (2)

k-particle Green functions interpreted as scattering amplitudes
purpose: Motivates higher characteristics; no physics derived
Unit 12 and footnote 12 explicitly call the physics motivation only, and no experimental prediction is made.
Hydrogen operator L-L^{-1}
purpose: Encodes total sphere Euler characteristic; nomenclature is formal analogy
Unit 8 introduces the operator and the Hydrogen analogy but no independent physical handle is provided.

how reviews work

0 comments

Cite this review

Pith. "Pith review of Elements of finite geometry I." pith.science (2026). https://pith.science/paper/G2B7G45O

@misc{pith2026260806405,
  author       = {Pith},
  title        = {Pith review of: Elements of finite geometry I},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/G2B7G45O}},
  note         = {Machine review of arXiv:2608.06405}
}

read the original abstract

This is a snapshot of a first part on a possibly much longer text on finite geometries, meaning graphs or finite abstract simplicial complexes. In in this first batch we review 12 subjects: Gauss-Bonnet, Poincare-Hopf, Index expectation, Euler's gem, Euler-Poincare,Unimodularity, Brouwer-Lefschetz, Sphere formula, Level sets, Index formula, Quadratic cohomology and Higher characteristic.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

202 extracted references · 50 canonical work pages

[1]

Aczel.Descartes’s secret notebook, a true tale of Mathematics, Mysticism and the Quest to Understand the Universe

A. Aczel.Descartes’s secret notebook, a true tale of Mathematics, Mysticism and the Quest to Understand the Universe. Broadway Books, 2005

2005
[2]

Alexandroff

P. Alexandroff. Diskrete R¨ aume.Mat. Sb. 2, 2, 1937

1937
[3]

Alexandroff and H

P. Alexandroff and H. Hopf.Topologie, volume 45 ofGrundlehren der Mathematischen Wis- senschaften. Springer, 1975

1975
[4]

Alexandrov.Combinatorial topology

P.S. Alexandrov.Combinatorial topology. Dover books on Mathematics. Dover Publications, Inc,
[5]

Allendoerfer and A

C. Allendoerfer and A. Weil. The Gauss-Bonnet theorem for Riemannian Polyhedra.Transac- tions of the American Mathematical Society, 53:101–129, 1943

1943
[6]

Allendoerfer

C.B. Allendoerfer. The Euler number of a Riemann manifold.Amer. J. Math., 62:243, 1940

1940
[7]

Balakrishnan and K

R. Balakrishnan and K. Ranganathan.A textbook of Graph Theory. Springer, 2012

2012
[8]

Banchoff

T. Banchoff. Critical points and curvature for embedded polyhedra.J. Differential Geometry, 1:245–256, 1967

1967

Show all 202 references

[9]

T. F. Banchoff. Critical points and curvature for embedded polyhedral surfaces.Amer. Math. Monthly, 77:475–485, 1970

1970
[10]

Berenstein, Enrico Casadio Tarabusi, Joel M

Carlos A. Berenstein, Enrico Casadio Tarabusi, Joel M. Cohen, and Massimo A. Picardello. Integral geometry on trees.Amer. J. Math., 113(3):441–470, 1991

1991
[11]

Springer Verlag, second edition, 1998

M.de Berg, M.van Kreveld, M.Overmars, and O.Schwarzkopf.Computational Geometry. Springer Verlag, second edition, 1998

1998
[12]

Berger.Jacob’s Ladder of Differential Geometry

M. Berger.Jacob’s Ladder of Differential Geometry. Springer Verlag, Berlin, 2009

2009
[13]

Biggs.Algebraic Graph Theory

N. Biggs.Algebraic Graph Theory. Cambridge University Press, 1974

1974
[14]

Bj¨ orner

A. Bj¨ orner. A cell complex in number theory.Advances in Appl. Math., 46:71–85, 2011

2011
[15]

Blaschke.Vorlesungen ¨ uber Integralgeometrie

W. Blaschke.Vorlesungen ¨ uber Integralgeometrie. Chelsea Publishing Company, New York, 1949

1949
[16]

Bobenko and Y

A. Bobenko and Y. Suris.Discrete Differential Geometry, Integrable Structure, volume 98 of Graduate Studies in Mathematics. AMS, 2008

2008
[17]

Bobenko, P

A.I. Bobenko, P. Schroeder, J.M. Sullivan, and B. Ziegler.Discrete Differential Geometry, vol- ume 38 ofOberwolfach Seminars. Birkhaueser, Basel, Boston, Berlin, 2008

2008
[18]

Bondy and U

J. Bondy and U. Murty.Graph theory, volume 244 ofGraduate Texts in Mathematics. Springer, New York, 2008

2008
[19]

Bowen and O

R. Bowen and O. E. Lanford, III. Zeta functions of restrictions of the shift transformation. In Global Analysis (Proc. Sympos. Pure Math., Vol. XIV, Berkeley, Calif., 1968), pages 43–49. Amer. Math. Soc., 1970

1968
[20]

Brenti and V

F. Brenti and V. Welker.f-vectors of barycentric subdivisions.Math. Z., 259(4):849–865, 2008

2008
[21]

Brown.The Lefschetz Fixed Point theorem

R.R. Brown.The Lefschetz Fixed Point theorem. Scott and Foresman, London, 1971

1971
[22]

Burde and H

G. Burde and H. Zieschang. Development of the concept of a complex. InHistory of Topology. Elsevier, 1999

1999
[23]

Chen, S-T

B. Chen, S-T. Yau, and Y-N. Yeh. Graph homotopy and Graham homotopy.Discrete Math., 241(1-3):153–170, 2001. Selected papers in honor of Helge Tverberg

2001
[24]

S.-S. Chern. A simple intrinsic proof of the Gauss-Bonnet formula for closed Riemannian man- ifolds.Annals of Mathematics, 45, 1944

1944
[25]

S-S. Chern. Historical remarks on Gauss-Bonnet. InAnalysis, et cetera, pages 209–217. Academic Press, Boston, MA, 1990

1990
[26]

Comwell.Polyhedra

P.R. Comwell.Polyhedra. Cambridge University Press, 1997

1997
[27]

Conway.Mathematical Connections: A Capstone Course

J.B. Conway.Mathematical Connections: A Capstone Course. American Mathematical Society, 2010

2010
[28]

Coxeter.Regular Polytopes

H.S.M. Coxeter.Regular Polytopes. Dover Publications, New York, 1973

1973
[29]

M. Crofton. On the theory of local probability. applied to straight lines drawn at random in a plane.Philosophical Transactions of the Royal Society of London, 158:181–199, 1968

1968
[30]

Cycon, R.G.Froese, W.Kirsch, and B.Simon.Schr¨ odinger Operators—with Application to Quantum Mechanics and Global Geometry

H.L. Cycon, R.G.Froese, W.Kirsch, and B.Simon.Schr¨ odinger Operators—with Application to Quantum Mechanics and Global Geometry. Springer-Verlag, 1987

1987
[31]

Dehn and P

M. Dehn and P. Heegaard. Analysis situs.Enzyklopaedie d. Math. Wiss, III.1.1:153–220, 1907. 58 OLIVER KNILL ELEMENTS OF FINITE GEOMETRY I

1907
[32]

Desbrun, E

M. Desbrun, E. Kanso, and Y. Tong. Discrete differential forms for computational modeling. In A. Bobenko, P. Schroeder, J. Sullivan, and G. Ziegler, editors,Discrete Differential Geometry, Oberwohlfach Seminars, 2008

2008
[33]

Desbrun, M

M. Desbrun, M. Leok, and J.E. Marsden. Discrete exterior calculus, preprint. http://arxiv.org/abs/math/0508341, 2003

2003 arXiv
[34]

Desbrun, M

M. Desbrun, M. Leok, and J.E. Marsden. Discrete Poincar´ e lemma.Appl. Numer. Math., 53(2- 4):231–248, 2005

2005
[35]

Desbrun and K

M. Desbrun and K. Polthier. Discrete differential geometry: An applied introduction. In Ari Stern Peter Schr¨ oder, Eitan Grinspun, editor,SIGGRAPH 2006, 2006

2006
[36]

Devadoss and J

S. Devadoss and J. O’Rourke.Discrete and Computational Geometry. Princeton University Press, 2011

2011
[37]

Diestel.Graph theory, volume 173 ofGraduate Texts in Mathematics

R. Diestel.Graph theory, volume 173 ofGraduate Texts in Mathematics. Springer, 5th edition, 2016

2016
[38]

Dieudonne.A History of Algebraic and Differential Topology, 1900-1960

J. Dieudonne.A History of Algebraic and Differential Topology, 1900-1960. Birkh¨ auser, 1989

1900
[39]

Eberhard.Morphologie der Polyeder

V. Eberhard.Morphologie der Polyeder. Teubner Verlag, 1891
[40]

B. Eckmann. Harmonische Funktionen und Randwertaufgaben in einem Komplex.Comment. Math. Helv., 17(1):240–255, 1944

1944
[41]

Edelsbrunner and J

H. Edelsbrunner and J. Harer.Computational topology, An introduction. AMS, Providence, RI, 2010

2010
[42]

Gowers (Editor).The Princeton Companion to Mathematics

T. Gowers (Editor).The Princeton Companion to Mathematics. Princeton University Press, 2008

2008
[43]

E. Engeler. An algorithmic model of strict finitism.Mathematical Logic in Computer Science, 26, 1978

1978
[44]

Esp ´ ınolaa and W.A

R. Esp ´ ınolaa and W.A. Krirkb. Fixed point theorems in r-trees with applications to graph theory.Topology and its Applications, 153:1046–1055, 2006

2006
[45]

W. Fenchel. On total curvatures for Riemannianm manifolds (i).J. London Math. Soc, 15:15, 1940

1940
[46]

Fomenko.Visual Geometry and Topology

A. Fomenko.Visual Geometry and Topology. Springer-Verlag, Berlin, 1994. From the Russian by Marianna V. Tsaplina

1994
[47]

R. Forman. A discrete Morse theory for cell complexes. InGeometry, topology, and physics, Conf. Proc. Lecture Notes Geom. Topology, IV, pages 112–125. Int. Press, Cambridge, MA, 1995

1995
[48]

R. Forman. Morse theory for cell complexes.Adv. Math., page 90, 1998

1998
[49]

R. Forman. Combinatorial differential topology and geometry.New Perspectives in Geometric Combinatorics, 38, 1999

1999
[50]

R. Forman. Bochner’s method for cell complexes and combinatorial ricci curvature.Discrete Comput. Geometry, pages 323–374, 2003

2003
[51]

J. Y. Gao. Topology in graph theory: A new perspective on classical concepts. Senior Thesis, Harvard University, 2024

2024
[52]

L. Glass. A combinatorial analog of the Poincare index theorem.Journal of combinatorial theory, 15:264–268, 1973

1973
[53]

Goedel.The consistency of the axiom of choice and of the generalized continuum hypothesis with the axioms of set theory

K. Goedel.The consistency of the axiom of choice and of the generalized continuum hypothesis with the axioms of set theory. Princeton University Press, 1940

1940
[54]

Granas and J

A. Granas and J. Dugundji.Fixed Point Theory. Springer Monographs in Mathematics. Springer Verlag, 2003

2003
[55]

Griffiths and J

P. Griffiths and J. Harris.Principles of Algebraic Geometry. Pure and Applied Mathematics. John Wiley and Sons, 1978

1978
[56]

Gr¨ unbaum

B. Gr¨ unbaum. Polytopes, graphs, and complexes.Bull. Amer. Math. Soc., 76:1131–1201, 1970

1970
[57]

Gr¨ unbaum

B. Gr¨ unbaum. Are your polyhedra the same as my polyhedra? InDiscrete and computational geometry, volume 25 ofAlgorithms Combin., pages 461–488. Springer, Berlin, 2003

2003
[58]

Gr¨ unbaum.Convex Polytopes

B. Gr¨ unbaum.Convex Polytopes. Springer, 2003

2003
[59]

Hadwiger.Vorlesungen ¨ uber Inhalt,Oberfl¨ ache und Isoperimetrie

H. Hadwiger.Vorlesungen ¨ uber Inhalt,Oberfl¨ ache und Isoperimetrie. Springer Verlag, Berlin, 1957, 1957

1957
[60]

Harary.Graph Theory

F. Harary.Graph Theory. Addison-Wesley Publishing Company, 1969. 59 ELEMENTS OF FINITE GEOMETRY I OLIVER KNILL

1969
[61]

Hatcher.Algebraic Topology

A. Hatcher.Algebraic Topology. Cambridge University Press, 2002

2002
[62]

G. Hetyei. The Stirling polynomial of a simplicial complex.Discrete and Computational Geom- etry, 35:437–455, 2006

2006
[63]

Y. Higuchi. Combinatorial curvature for planar graphs.J. Graph Theory, 38:220–229, 2001

2001
[64]

D. R. Hofstadter.Goedel, Escher Bach, an eternal golden braid. Basic Books, 1979

1979
[65]

H. Hopf. ¨Uber die Curvatura integra geschlossener Hyperfl¨ achen.Math. Ann., 95(1):340–367, 1926

1926
[66]

H. Hopf. ¨Uber die Curvatura integra geschlossener Hyperflaechen.Mathematische Annalen, 95:340–367, 1926

1926
[67]

H. Hopf. A new proof of the Lefschetz formula on invariant points.Proc. Nat. Acad. Sci., 14:149– 153, 1928

1928
[68]

Horak and J

D. Horak and J. Jost. Spectra of combinatorial Laplace operators on simplicial complexes.Adv. Math., 244:303–336, 2013

2013
[69]

A. V. Ivashchenko. Representation of smooth surfaces by graphs. transformations of graphs which do not change the euler characteristic of graphs.Discrete Math., 122:219–233, 1993

1993
[70]

Ivashchenko

A.V. Ivashchenko. Graphs of spheres and tori.Discrete Math., 128(1-3):247–255, 1994

1994
[71]

J. James. History of topology. InHistory of Topology, 1999

1999
[72]

Jezierski and W

J. Jezierski and W. Marzantowicz.Homotopy methods in topological fixed and periodic points theory. Springer Verlag, 2006

2006
[73]

Goodman-Strauss J.H

C. Goodman-Strauss J.H. Conway, H.Burgiel.The Symmetries of Things. A.K. Peterse, Ltd., 2008

2008
[74]

Jonsson.Simplicial Complexes of Graphs, volume 1928 ofLecture Notes in Mathematics

J. Jonsson.Simplicial Complexes of Graphs, volume 1928 ofLecture Notes in Mathematics. Springer, 2008

1928
[75]

Josellis and O

F. Josellis and O. Knill. A Lusternik-Schnirelmann theorem for graphs. http://arxiv.org/abs/1211.0750, 2012

2012 arXiv
[76]

Jost and S

J. Jost and S. Liu. Ollivier’s Ricci curvature, local clustering and curvature dimension inequalities on graphs. arXiv:1103.4037v2, April 1, 2011, 2011

2011 arXiv
[77]

R.M. Karp. Reducibility among combinatorial problems. In In R. E. Miller and J. W. Thatcher, editors,Complexity of Computer Computations, pages 85–103. Plenum, 1972

1972
[78]

Kirchhoff

G. Kirchhoff. ¨Uber die Aufl¨ osung der Gleichungen auf welche man bei der Untersuchung der linearen Verteilung galvanischer Str¨ ome gef¨ uhrt wird.Ann. Phys. Chem., 72:497–508, 1847
[79]

D. Klain. Dehn-Sommerville relations for triangulated manifolds. http://faculty.uml.edu/dklain/ds.pdf, 2002

2002
[80]

D.A. Klain. A short proof of hadwiger’s characterization theorem.Mathematika, 42:329–339, 95
[81]

Klain and G-C

D.A. Klain and G-C. Rota.Introduction to geometric probability. Lezioni Lincee. Accademia nazionale dei lincei, 1997

1997
[82]

V. Klee. The Euler characteristic in combinatorial geometry.The American Mathematical Monthly, 70(2):pp. 119–127, 1963

1963
[83]

V. Klee. A combinatorial analogue of Poincar´ e’s duality theorem.Canadian J. Math., 16:517– 531, 1964

1964
[84]

O. Knill. The upper Lyapunov exponent of SL(2,R) cocycles: discontinuity and the problem of positivity. InLyapunov exponents (Oberwolfach, 1990), volume 1486 ofLecture Notes in Math., pages 86–97. Springer, Berlin, 1991

1990
[85]

O. Knill. Positive lyapunov exponents for a dense set of bounded measurable sl(2,r)-cocycles. Ergodic Theory and Dynamical Systems, 12(2):319–331, 1992

1992
[86]

O. Knill. A graph theoretical Gauss-Bonnet-Chern theorem. http://arxiv.org/abs/1111.5395, 2011

2011 arXiv
[87]

O. Knill. A discrete Gauss-Bonnet type theorem.Elemente der Mathematik, 67:1–17, 2012

2012
[88]

O. Knill. A graph theoretical Poincar´ e-Hopf theorem. http://arxiv.org/abs/1201.1162, 2012

2012 arXiv
[89]

O. Knill. An index formula for simple graphs . http://arxiv.org/abs/1205.0306, 2012

2012 arXiv
[90]

O. Knill. On index expectation and curvature for networks. http://arxiv.org/abs/1202.4514, 2012. 60 OLIVER KNILL ELEMENTS OF FINITE GEOMETRY I

2012 arXiv
[91]

O. Knill. The McKean-Singer Formula in Graph Theory. http://arxiv.org/abs/1301.1408, 2012

2012 arXiv
[92]

O. Knill. The theorems of Green-Stokes,Gauss-Bonnet and Poincare-Hopf in Graph Theory. http://arxiv.org/abs/1201.6049, 2012

2012 arXiv
[93]

O. Knill. A Brouwer fixed point theorem for graph endomorphisms.Fixed Point Th. and Appl., 85, 2013

2013
[94]

O. Knill. The Dirac operator of a graph. http://arxiv.org/abs/1306.2166, 2013

2013 arXiv
[95]

O. Knill. The Euler characteristic of an even-dimensional graph. http://arxiv.org/abs/1307.3809, 2013

2013 arXiv
[96]

O. Knill. Classical mathematical structures within topological graph theory. http://arxiv.org/abs/1402.2029, 2014

2014 arXiv
[97]

O. Knill. If Archimedes would have known functions . http://arxiv.org/abs/1403.5821, 2014

2014 arXiv
[98]

O. Knill. Golden rotations. Talk at BU of Februaray 23, 2015, 2015

2015
[99]

O. Knill. A Sard theorem for graph theory. http://arxiv.org/abs/1508.05657, 2015

2015 arXiv
[100]

O. Knill. Bowen lanford zeta function. math table talk on 10/18/2016, 2016

2016
[101]

O. Knill. Gauss-Bonnet for multi-linear valuations. http://arxiv.org/abs/1601.04533, 2016

2016 arXiv
[102]

O. Knill. On Fredholm determinants in topology. https://arxiv.org/abs/1612.08229, 2016

2016 arXiv
[103]

O. Knill. On primes, graphs and cohomology. https://arxiv.org/abs/1608.06877, 2016

2016 arXiv
[104]

O. Knill. Some experiments in number theory. https://arxiv.org/abs/1606.05971, 2016

2016 arXiv
[105]

O. Knill. Wu characteristic. Math table talk on 3/8/2016, 2016

2016
[106]

O. Knill. The cohomology for Wu characteristics. http://arxiv.org/abs/1803.06788, 2017

2017 arXiv
[107]

O. Knill. On a Dehn-Sommerville functional for simplicial complexes. https://arxiv.org/abs/1705.10439, 2017

2017 arXiv
[108]

O. Knill. The amazing world of simplicial complexes. https://arxiv.org/abs/1804.08211, 2018

2018 arXiv
[109]

O. Knill. Polishing euler’s gem. math table talk on 2/6/18 before Euler’s day 2/7/18, 2018

2018
[110]

O. Knill. The average simplex cardinality of a finite abstract simplicial complex. https://arxiv.org/abs/1905.02118, 2019

1905 arXiv
[111]

O. Knill. Constant index expectation curvature for graphs or Riemannian manifolds. https://arxiv.org/abs/1912.11315, 2019

1912 arXiv
[112]

O. Knill. The counting matrix of a simplicial complex. https://arxiv.org/abs/1907.09092, 2019

1907 arXiv
[113]

O. Knill. Dehn-Sommerville from Gauss-Bonnet. https://arxiv.org/abs/1905.04831, 2019

1905 arXiv
[114]

O. Knill. Energized simplicial complexes. https://arxiv.org/abs/1908.06563, 2019

1908 arXiv
[115]

O. Knill. More on Poincar´ e-Hopf and Gauss-Bonnet. https://arxiv.org/abs/1912.00577, 2019

1912 arXiv
[116]

O. Knill. A parametrized Poincare-Hopf theorem and clique cardinalities of graphs. https://arxiv.org/abs/1906.06611, 2019

1906 arXiv
[117]

O. Knill. Poincar´ e-Hopf for vector fields on graphs. https://arxiv.org/abs/1911.04208, 2019

1911 arXiv
[118]

O. Knill. A Reeb sphere theorem in graph theory. https://arxiv.org/abs/1903.10105, 2019

1903 arXiv
[119]

O. Knill. Complexes, Graphs, Homotopy, Products and Shannon Capacity. https://arxiv.org/abs/2012.07247, 2020

2012 arXiv
[120]

O. Knill. Division algebra valued energized simplicial complexes. https://arxiv.org/abs/2008.10176, 2020

2008 arXiv
[121]

O. Knill. The energy of a simplicial complex.Linear Algebra and its Applications, 600:96–129, 2020

2020
[123]

O. Knill. Green functions of energized complexes. https://arxiv.org/abs/2010.09152, 2020

2010 arXiv
[124]

O. Knill. Integral geometric Hopf conjectures. https://arxiv.org/abs/2001.01398, 2020

2001 arXiv
[125]

O. Knill. On index expectation curvature for manifolds. https://arxiv.org/abs/2001.06925, 2020

2001 arXiv
[126]

O. Knill. Characteristic topological invariants. https://arxiv.org/abs/2302.02510, 2023

2023 arXiv
[127]

O. Knill. Cohomology of open sets, 2023. https://arxiv.org/abs/2305.12613

2023 arXiv
[128]

O. Knill. Finite topologies for finite geometries. http://arxiv.org/abs/2301.03156, 2023

2023 arXiv
[129]

O. Knill. The sphere formula. https://arxiv.org/abs/2301.05736, 2023

2023 arXiv
[130]

O. Knill. Fusion inequality for quadratic cohomology, 2024

2024
[131]

O. Knill. Gauss-Bonnet for Form Curvatures. https://arxiv.org/abs/2409.01425, 2024

2024 arXiv
[132]

O. Knill. Manifolds from partitions. https://arxiv.org/abs/2401.07435, 2024

2024 arXiv
[133]

O. Knill. Morse and Lusternik-Schnirelmann for graphs. https://arxiv.org/abs/2405.19603, 2024

2024 arXiv
[134]

O. Knill. Dehn sommerville manifolds. https://arxiv.org/abs/2508.14372, 2025

2025 arXiv
[135]

O. Knill. Block Jacobi matrices, Barycentric limits and Manifolds. https://arxiv.org/abs/2601.10815, 2026

2026
[136]

O. Knill. Remarks about Connection and Dirac matrices. https://arxiv.org/abs/2601.18071, 2026

2026 arXiv
[137]

Knill and T

O. Knill and T. Tucker. A Riemann-Hurwitz theorem in graph theory. Draft Notes, December 2012

2012
[138]

H. W. Kuhn and S. Nasar.The Essential Nash. Princeton University Press, 2002

2002
[139]

Kustaanheimo

P. Kustaanheimo. On the fundamental prime of a finite world.Ann. Acad. Sci. Fennicae. Ser. A. I. Math.-Phys., 1952(129):7, 1952

1952
[140]

Lakatos.Proofs and Refutations

I. Lakatos.Proofs and Refutations. Cambridge University Press, 1976

1976
[141]

Lanford, III

Oscar E. Lanford, III. Informal remarks on the orbit structure of discrete approximations to chaotic maps.Experiment. Math., 7(4):317–324, 1998

1998
[142]

Lanford III

O. Lanford III. Introduction to hyperbolic sets. In A.S.Wightman G.Velo, editor,Regular and Chaotic Motion in Dynamical systems, volume 118 ofNato ASI Ser. B, pages 73–102. Plenum Press, 1985

1985
[143]

Lefschetz

S. Lefschetz. Intersections and transformations of complexes and manifolds.Trans. Am. Math.Soc, 28:1–49, 1926

1926
[144]

N. Levitt. The Euler characteristic is the unique locally determined numerical homotopy invari- ant of finite complexes.Discrete Comput. Geom., 7:59–67, 1992

1992
[145]

Luzon and M.A

A. Luzon and M.A. Moron. Pascal triangle, Stirling numbers and the unique invariance of the euler characteristic. arxiv.1202.0663, 2012

2012 arXiv
[146]

Manolescu

C. Manolescu. Triangulations of manifolds. InNotices of the International Congress of Chinese Mathematicians, volume 2, pages 21–23, 2014

2014
[147]

May.Simplicial Objects in Algebraic Topology

J.P. May.Simplicial Objects in Algebraic Topology. Chicago lectures in Mathematics. University of Chicago Press, 1967

1967
[148]

J.P. May. Finite topological spaces. Notes for REU, Chicago, 2003-2008, 2008

2003
[149]

McKean and I.M

H.P. McKean and I.M. Singer. Curvature and the eigenvalues of the Laplacian.J. Differential Geometry, 1(1):43–69, 1967

1967
[150]

Merris.Graph theory

R. Merris.Graph theory. Interscience Series in discrete mathematics and optimization. Wiley, 2001

2001
[151]

A.P. Morse. The behavior of a function on its critical set.Ann. of Math. (2), 40(1):62–70, 1939

1939
[152]

Mukherjee and S

S.K. Mukherjee and S. Bera. A simple elementary proof of The Unimodularity Theorem of Oliver Knill.Linear Algebra and Its applications, pages 124–127, 2018

2018
[153]

Mullen, A

P. Mullen, A. McKenzie, D. Pavlov, L. Durant, Y. Tong, E. Kanso, J. E. Marsden, and M. Des- brun. Discrete Lie advection of differential forms.Found. Comput. Math., 11(2):131–149, 2011

2011
[154]

Munkholm and H.J

E.S. Munkholm and H.J. Munkholm. Poul Heegaard, The Dehn-Heegaard Enzyklop¨ adie arti- cle (1907). http://www.imada.sdu.dk/ ˜ hjm/heegaard3.stor/heegaard3.stor.html, 1998, Accessed, October 4, 2017

1907
[155]

Murai and I

S. Murai and I. Novik. Face numbers of manifolds with boundary. http://arxiv.org/abs/1509.05115, 2015. 62 OLIVER KNILL ELEMENTS OF FINITE GEOMETRY I

2015 arXiv
[156]

E. Nelson. Internal set theory: A new approach to nonstandard analysis.Bull. Amer. Math. Soc, 83:1165–1198, 1977

1977
[157]

Nelson.Radically elementary probability theory

E. Nelson.Radically elementary probability theory. Princeton university text, 1987

1987
[158]

Nicolaescu.Lectures on the Geometry of Manifolds

L. Nicolaescu.Lectures on the Geometry of Manifolds. World Scientific, second edition, 2009

2009
[159]

Novik and E

I. Novik and E. Swartz. Applications of Klee’s Dehn-Sommerville relations.Discrete Comput. Geom., 42(2):261–276, 2009

2009
[160]

Nowakowski and I

R. Nowakowski and I. Rival. Fixed-edge theorem for graphs with loops.J. Graph Theory, 3:339– 350, 1979

1979
[161]

Ollivier

Y. Ollivier. Ricci curvature of Markov chains on metric spaces.J. Funct. Anal., 256:810–864, 2009

2009
[162]

Rieben P

R. Rieben P. Castilo, J. Koning and D. White. A discrete differential forms framework for computational electromagnetism.CMES, 1:1–15, 2002

2002
[163]

Goodman P

A.W. Goodman P. Erd¨ os and L. P´ osa. The respresentation of a graph by a set intersections. Canadian Journal of Mathematics, 18:106–112, 1966

1966
[164]

K. Perkins. Discrete analogs of the poincar´ e hopf theorem. Senior thesis Harvey Mudd college, 2023

2023
[165]

Poincar´ e

H. Poincar´ e. Sur les courbes definies par les equation differentielle III.Journal de Mathematique pures et appliqu´ ees, pages 167–244, 1885
[166]

Polo-Blanco

I. Polo-Blanco. Alicia Boole Stott, a geometer in higher dimension.Historia Mathematica, 35(2):123 – 139, 2008

2008
[167]

C. Pugh. A generalized Poincar´ e index formula.Topology, 7:217–226, 1968

1968
[168]

F. Rannou. Numerical study of discrete area-preserving mappings.Acta Arithm, 31:289–301, 1974

1974
[169]

T. Regge. General relativity without coordinates.Nuovo Cimento (10), 19:558–571, 1961

1961
[170]

Rhodes and P.V

J. Rhodes and P.V. Silva.Boolean Representations of Simplicial Complexes and Matroids. Springer Monographs in Mathematics, 2015

2015
[171]

Richeson.Euler’s Gem

D.S. Richeson.Euler’s Gem. Princeton University Press, Princeton, NJ, 2008. The polyhedron formula and the birth of topology

2008
[172]

Robert.Analyse non standard

A. Robert.Analyse non standard. Presses polytechniques romandes, 1985

1985
[173]

G-C. Rota. On the combinatorics of the Euler characteristic. InStudies in Pure Mathematics (Presented to Richard Rado), pages 221–233. Academic Press, London, 1971

1971
[174]

Rotman.An introduction to Algebraic Topology

J.J. Rotman.An introduction to Algebraic Topology. Graduate Texts in Mathematics. Springer
[175]

Santalo.Introduction to integral geometry

L.A. Santalo.Introduction to integral geometry. Hermann and Editeurs, Paris, 1953

1953
[176]

Santalo.Integral Geometry and Geometric Probability

L.A. Santalo.Integral Geometry and Geometric Probability. Cambridge University Press, second edition, 2004

2004
[177]

A. Sard. The measure of the critical values of differentiable maps.Bull. Amer. Math. Soc., 48:883–890, 1942

1942
[178]

A. Sard. Images of critical sets.Ann. of Math. (2), 68:247–259, 1958

1958
[179]

Schl¨ afli.Theorie der Vielfachen Kontinuit¨ at

L. Schl¨ afli.Theorie der Vielfachen Kontinuit¨ at. Cornell University Library Digital Collections, 1901

1901
[180]

Schneider.Integral geometric tools for stochastic geometry, volume 1892 ofLecture Notes in Math.Springer, 2007

R. Schneider.Integral geometric tools for stochastic geometry, volume 1892 ofLecture Notes in Math.Springer, 2007

2007
[181]

E. Scholz. The concept of manifold, 1850-1950. InHistory of Topology. Elsevier, 1999

1950
[182]

Schoute.Analytical treatment of the polytopes regularly derived from the regular polytopes

P.H. Schoute.Analytical treatment of the polytopes regularly derived from the regular polytopes. Johannes Mueller, 1911

1911
[183]

C. Shannon. The zero error capacity of a noisy channel.IRE Transactions on Information Theory, 2:8–19, 1956

1956
[184]

Spanier.Algebraic Topology

E.H. Spanier.Algebraic Topology. Springer Verlag, 1966

1966
[185]

Spivak.A comprehensive Introduction to Differential Geometry V

M. Spivak.A comprehensive Introduction to Differential Geometry V. Publish or Perish, Inc, Berkeley, third edition, 1999

1999
[186]

Stanley.Enumerative Combinatorics, Vol

R. Stanley.Enumerative Combinatorics, Vol. I. Wadworth and Brooks/Cole, 1986

1986
[187]

Stanley.Combinatorics and Commutative Algebra

R. Stanley.Combinatorics and Commutative Algebra. Progress in Math. Birkh¨ auser, second edition, 1996

1996
[188]

Sutton.Platonic and Archimedean Solids

D. Sutton.Platonic and Archimedean Solids. Walker and Company, New York, 2002. 63 ELEMENTS OF FINITE GEOMETRY I OLIVER KNILL

2002
[189]

Szpilrajn-Marczewski

E. Szpilrajn-Marczewski. Sur deux propriet´ es des classes d’ensembles.Fund. Math., 33:303–307,
[190]

van der Waerden.Algebra, volume 1

B.L. van der Waerden.Algebra, volume 1. Springer, New York, 1991

1991
[191]

Veblen.Analysis Situs

O. Veblen.Analysis Situs. The Cambridge Colloqium lectures volume V. AMS, 1922

1922
[192]

Topological invariants of new type of finite polyhedrons.Acta Math

Wu W-T. Topological invariants of new type of finite polyhedrons.Acta Math. Sinica, 3:261–290, 1953

1953
[193]

Weyl.Riemanns geometrische Ideen, ihre Auswirkung und ihre Verkn¨ upfung mit der Grup- pentheorie

H. Weyl.Riemanns geometrische Ideen, ihre Auswirkung und ihre Verkn¨ upfung mit der Grup- pentheorie. Springer Verlag, 1925, republished 1988

1925
[194]

Whitehead

J.H.C. Whitehead. Simplicial spaces, nuclei and m-groups.Proc. London Math. Soc., 45(1):243– 327, 1939

1939
[195]

Whitney.Geometric integration theory

H. Whitney.Geometric integration theory. Princeton University Press, Princeton, N. J., 1957

1957
[196]

H. Wu. Historical development of the Gauss-Bonnet theorem.Science in China Series A, 2007

2007
[197]

J-C. Yoccoz. Introduction to hyperbolic dynamics. InReal and complex dynamical systems (Hillerød, 1993), volume 464 ofNATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., pages 265–

1993
[198]

E.C. Zeeman. On the dunce hat.Topology, 2:341–358, 1964

1964
[199]

Ziegler.Lectures on Polytopes

G.M. Ziegler.Lectures on Polytopes. Springer Verlag, 1995

1995
[200]

A.A. Zykov. On some properties of linear complexes. (Russian).Mat. Sbornik N.S., 24(66):163– 188, 1949. Oliver Knill, knill@math.harvard.edu, Elements of Finite Geometry I, 2026 64

1949
[291]

Publ., Dordrecht, 1995

Kluwer Acad. Publ., Dordrecht, 1995

1995
[1945]

Burlingham and L

Translation by B. Burlingham and L. Stewart, 2009

2009
[1956]

Three volumes bound as one

Pith tools

Reviewed August 10, 2026 · model on record in the stance chip above.

[1] [1]

Aczel.Descartes’s secret notebook, a true tale of Mathematics, Mysticism and the Quest to Understand the Universe

A. Aczel.Descartes’s secret notebook, a true tale of Mathematics, Mysticism and the Quest to Understand the Universe. Broadway Books, 2005

2005

[2] [2]

Alexandroff

P. Alexandroff. Diskrete R¨ aume.Mat. Sb. 2, 2, 1937

1937

[3] [3]

Alexandroff and H

P. Alexandroff and H. Hopf.Topologie, volume 45 ofGrundlehren der Mathematischen Wis- senschaften. Springer, 1975

1975

[4] [4]

Alexandrov.Combinatorial topology

P.S. Alexandrov.Combinatorial topology. Dover books on Mathematics. Dover Publications, Inc,

[5] [5]

Allendoerfer and A

C. Allendoerfer and A. Weil. The Gauss-Bonnet theorem for Riemannian Polyhedra.Transac- tions of the American Mathematical Society, 53:101–129, 1943

1943

[6] [6]

Allendoerfer

C.B. Allendoerfer. The Euler number of a Riemann manifold.Amer. J. Math., 62:243, 1940

1940

[7] [7]

Balakrishnan and K

R. Balakrishnan and K. Ranganathan.A textbook of Graph Theory. Springer, 2012

2012

[8] [8]

Banchoff

T. Banchoff. Critical points and curvature for embedded polyhedra.J. Differential Geometry, 1:245–256, 1967

1967