" I divide mathematics into discrete and continuous
(prickles and goo, as Alan Watts put it) . . . ."
— Peter J. Cameron on 8 December 2024
"What is a GUI?" —
See also an illustration from "Google's Apple Tree" (Jan. 4, 2010) —
* Title purloined from Gian-Carlo Rota.
FutureHouse diamond theorem report on May 3, 2025 —
From https://platform.futurehouse.org/trajectories/995a20ed-9de3-43b9-83d2-64d4b8b92bf2.
What is the Cullinane diamond theorem?
This report presents a detailed exposition of the Cullinane diamond theorem, a result that interweaves combinatorics, finite projective geometry, symmetry, and algebra. In its original formulation, the theorem concerns a 4×4 array composed of 16 two‐color, diagonally divided square tiles—a configuration referred to as the diamond figure D—and describes the remarkable invariance properties of D when acted on by a very large permutation group. This report explains the theorem’s statement, the group–theoretic framework underlying it, its geometric interpretation in terms of finite projective spaces, and its algebraic extensions via delta transforms. In doing so, the discussion also elucidates connections with Latin square geometry and deep algebraic structures such as those underlying the Miracle Octad Generator (MOG) and Mathieu groups.
Abstract
The Cullinane diamond theorem asserts that every image of the four‐diamond figure D, arising from a 4×4 two‐color diagonally divided square array, exhibits either ordinary or color–interchange symmetry under a group G of order 322,560. (cullinane2013thediamondtheorem pages 1-5) This transformation group, generated by arbitrary permutations of the rows, columns, and the four 2×2 quadrants of the array, is isomorphic to the affine group acting on a 4–dimensional vector space over the finite field GF(2). The theorem further establishes that the 840 distinct images of D decompose naturally into 35 equivalence classes—each corresponding bijectively to one of the 35 lines in the three–dimensional projective space PG(3,2). In each equivalence class the images can be represented by three interrelated line diagrams whose entries, under binary (mod 2) addition, sum to zero. This binary-addition property not only reflects the underlying structure of PG(3,2) but also serves as a combinatorial invariant that connects the pattern symmetries with the algebra of finite fields. (cullinane2013thediamondtheorem pages 1-5) Moreover, by considering the so-called delta transforms on arrays—where each element of a square array is replaced by a uniquely determined diagonally divided two–color tile—an ideal is produced within a larger ring of symmetric patterns. Such an ideal, consisting in one instance of 1024 “diamond” patterns within a ring of 4096 symmetric configurations, paves the way for an infinite family of “diamond” rings that are isomorphic to matrix rings over GF(4). (cullinane2013thediamondtheorem pages 1-5, cullinaneUnknownyearexamples pages 1-1) In addition, the symmetry group involved in the theorem is intimately related to the octad stabilizer subgroup within the Mathieu group M24, as emphasized in studies of the Miracle Octad Generator. (cullinane2013thediamondtheorem pages 1-5, kellyUnknownyearmathieugroupsthe pages 1-1)
The Cullinane diamond theorem occupies a position of central importance in several overlapping domains of mathematics. Its beauty lies in how a deceptively simple graphic design—the four–diamond figure D obtained from a 4×4 array of specially divided square tiles—encodes deep symmetry properties when subjected to highly structured group actions. The theorem was originally developed to provide a purely geometric explanation for longstanding puzzles in symmetric pattern design, yet its ramifications extend to Latin square theory, coding theory, and even computer–aided secret sharing in cryptography. (cullinane2013thediamondtheorem pages 1-5) By using group actions derived from the affine group over GF(2), Cullinane demonstrated that the resulting images not only preserve symmetry but also organize themselves in a manner that reflects the structure of the finite projective space PG(3,2). This report systematically outlines the theorem, providing the necessary mathematical background and exploring its broader significance.
At the heart of the theorem is the diamond figure D—a 4×4 array whose 16 unit squares are each divided along a diagonal into two contrasting colors. This design is not arbitrary; it is constructed so that when transformations are applied, its inherent symmetry properties become evident. The large permutation group G, of order 322,560, is generated by all possible permutations of the rows, the columns, and the four 2×2 quadrants. (cullinane2013thediamondtheorem pages 1-5) An essential observation is that G is isomorphic to the full affine group on a four–dimensional vector space over GF(2), where GF(2) is the finite field with two elements. The affine structure imparts a rich algebraic framework that facilitates rigorous combinatorial analysis. Each element of G rearranges the tiles of D, yet—remarkably—the resulting pattern always exhibits a precise form of symmetry, be it an ordinary symmetry (a geometric transformation mapping the pattern to itself) or a color–interchange symmetry (where interchanging the two colors yields an invariant image).
One of the most striking outcomes of Cullinane’s work is the enumeration of the distinct images of D under the action of G. Detailed analysis reveals that there are exactly 840 such images. These 840 images do not form a homogeneous collection; instead, they naturally partition into 35 distinct equivalence classes. (cullinane2013thediamondtheorem pages 1-5) This partitioning is not coincidental. In fact, there is a bijective correspondence between the 35 equivalence classes of images and the 35 lines in PG(3,2)—the projective space of dimension three over GF(2). In finite projective geometry, PG(3,2) is a highly symmetric structure that contains 15 points and 35 lines, and the incidence relations among these geometric subspaces mirror the combinatorial relationships found among the images of D. Thus, the combinatorial arrangement of tiles in D under all G–images embodies a finite geometric structure that is isomorphic to PG(3,2). (cullinane2013thediamondtheorem pages 1-5)
Each of the 35 equivalence classes can be concretely visualized via collections of three interrelated diagrams known as line diagrams. These diagrams are so constructed that, when added together modulo 2 (i.e., performing binary addition on their entries), the resulting sum is zero. This property is highly significant; it encapsulates the idea that the three diagrams represent three distinct partitions of the four tiles into two subsets, and the symmetry is maintained by the fact that their binary sum (in the field GF(2)) vanishes. (cullinane2013thediamondtheorem pages 1-5) In effect, the line diagrams serve as a pictorial and algebraic manifestation of the structure of PG(3,2). The binary-addition condition is reminiscent of the behavior of vectors in a finite vector space, reinforcing the interpretation of the underlying symmetries in linear algebraic terms. This representation is of particular interest in algebraic combinatorics, as it provides a concrete invariant that can be used to classify and analyze symmetric patterns generated by G.
Beyond the geometric interpretation lies a powerful algebraic generalization. The theorem has been extended by considering “delta transforms” of square arrays. A delta transform is defined as a one-to-one substitution procedure in which each entry of an array (often arising from a Latin square or a similar combinatorial object) is replaced by a fixed diamond pattern—a diagonally divided, two–colored unit square. (cullinaneUnknownyearexamples pages 1-1) When applied to structured arrays such as the Klein group table (which itself can be viewed as a Latin square over GF(4)), the delta transform preserves the symmetry properties inherent in the original configuration. This invariance under delta transforms implies that the entire algebra generated by the images of the Klein group table under G comprises solely symmetrical arrays. More precisely, these images generate an ideal in a larger ring—a ring of 4096 symmetric patterns—from which one can extract an ideal consisting of 1024 “diamond” patterns. The algebraic structure revealed in this manner is so robust that it generalizes to an infinite family of diamond rings, each of which is isomorphic to a matrix ring over GF(4). (cullinane2013thediamondtheorem pages 1-5, cullinaneUnknownyearexamples pages 1-1) This connection to matrix rings over finite fields accentuates the deep interplay between combinatorial design and algebraic structures.
Another fascinating aspect of the Cullinane diamond theorem is its relation to Latin square geometry—a classical topic in combinatorics that deals with square arrays in which each symbol occurs exactly once per row and once per column. In some of Cullinane’s later work, particularly in his study of Latin-square geometry, it is shown that the six 4×4 Latin squares (that have orthogonal Latin mates) can be embedded into a set of 35 arrays in a manner that mirrors the correspondence between the diamond images and the 35 lines of PG(3,2). (cullinaneUnknownyearlatinsquaregeometry pages 1-6) In this interpretation, the orthogonality property of Latin squares is translated into a geometric condition: two Latin squares are orthogonal if and only if the corresponding lines in PG(3,2) are skew (that is, they do not intersect). This geometric visualization not only provides intuition for the phenomenon of orthogonality but also serves as an explicit bridge between classical combinatorial design and finite projective geometry. In doing so, it enriches our understanding of both domains while demonstrating the versatility of the diamond theorem’s underlying principles.
The permutation group G, with its staggering order of 322,560, is by itself an object of intense interest in group theory. Much more than a tool for rearranging tiles, G is isomorphic to the affine group acting on the 4-dimensional linear space over GF(2). This same group appears elsewhere in mathematics, in particular as the octad stabilizer in the Mathieu group M24, a sporadic simple group that plays a central role in combinatorial design and coding theory. In fact, R. T. Curtis’s Miracle Octad Generator (MOG)—developed as a way to generate and study the Golay code (an exceptional error–correcting code) and related combinatorial structures—utilizes a configuration strongly reminiscent of the diamond–theorem figures. (cullinane2013thediamondtheorem pages 1-5, kellyUnknownyearmathieugroupsthe pages 1-1) This correspondence highlights the deep algebraic and combinatorial unity underlying what might initially appear as unrelated phenomena: the design of quilt patterns and the structure of error–correcting codes.
To appreciate the full depth of the Cullinane diamond theorem, it is instructive to examine the group–theoretic foundations in greater detail. The generator set for the group G comprises three independent types of permutations—those acting on rows, on columns, and on the four 2×2 quadrants. This decomposition implies that every element of G can be represented as a combination of three distinct permutations, each contributing to the overall transformation of the array D. When these permutations are interpreted within the framework of an affine vector space over GF(2), one observes that their composition corresponds to linear transformations accompanied by translations. (cullinane2013thediamondtheorem pages 1-5) This realization not only explains why G is isomorphic to an affine group but also establishes a link between the combinatorial structure of the tiled array and the rich theory of finite fields and linear algebra. Such a connection is essential to both the formulation and the proof of the theorem.
The finite field GF(2) consists of just two elements—0 and 1—which endow any vector space over GF(2) with a binary structure. In the context of the diamond theorem, every tile’s coloring, as well as the additive relations in the line diagrams, are naturally described by elements of GF(2). Moreover, the projective space PG(3,2) arises from considering the nonzero vectors in the four–dimensional space over GF(2) up to scalar multiples. PG(3,2) contains exactly 15 points and 35 lines; it is precisely this enumeration of lines that inspires the classification of the 840 images of D into 35 equivalence classes. (cullinane2013thediamondtheorem pages 1-5) The binary addition (mod 2) property of the three line diagrams representing each class mirrors the fact that, in PG(3,2), any three collinear points obey a linear relation summing to zero. This elegant correspondence between abstract finite geometry and the tangible patterns of a tiled array is one of the most striking features of the theorem.
An additional layer of sophistication in the theorem’s framework is provided by the concept of delta transforms. A delta transform is a systematic substitution process in which every entry of a square array (often drawn from a four–element set) is replaced by a fixed, diagonally divided two–colored tile. (cullinaneUnknownyearexamples pages 1-1) When Delta transforms are applied to the table corresponding to the Klein group, the resulting new arrays (called delta transforms of the Klein group table) retain either ordinary symmetry or color–interchange symmetry. This invariance is maintained under the full group G, which means that the delta transform itself is an operation that commutes with the action of G. The combinatorial invariant arising from the delta transforms is highly significant because it allows one to define sums and products on the set of G–images of D, thereby generating a ring of symmetric patterns. In particular, this ring contains an ideal consisting of 1024 diamond patterns and generalizes to an infinite family of diamond rings isomorphic to matrix rings over GF(4). (cullinane2013thediamondtheorem pages 1-5, cullinaneUnknownyearexamples pages 1-1) The elegance of this result lies in the seamless transition from a discrete combinatorial construct to a rich algebraic structure.
The principles behind the Cullinane diamond theorem have further inspired research into Latin square geometry. In the special case of 4×4 Latin squares, it has been shown that the six Latin squares possessing orthogonal Latin mates can be embedded within a configuration of 35 arrays. (cullinaneUnknownyearlatinsquaregeometry pages 1-6) In this embedding, the traditional notion of orthogonality of Latin squares—originally based on combinatorial criteria—corresponds exactly to the geometric property of skewness (i.e., the non–intersection of lines) in the projective space PG(3,2). This geometric interpretation offers not only a new perspective on the classical problem of constructing mutually orthogonal Latin squares but also demonstrates that the diamond theorem’s reach extends well beyond its original domain of tiling patterns. The correspondence essentially means that the combinatorial structure of a Latin square is mirrored in the arrangement of lines in a finite projective space, and the multiple representations provided by the delta transform further cement this connection. (cullinaneUnknownyearlatinsquaregeometry pages 1-6) This interplay between Latin square geometry and finite projective spaces opens up opportunities for deeper exploration of geometrical invariants and symmetric designs.
[ Correction by Cullinane on June 11, 2025 – This section is in error and should be ignored. ]
While the Cullinane diamond theorem is rooted in abstract combinatorial and geometric concepts, its influence extends into various applied fields. In the domain of facility location, for example, researchers have exploited similar “diamond” structures to characterize regions where optimal locations occur under the rectilinear (L1) norm, as these regions naturally form diamond–shaped loci defined by distance constraints. (giannikos1993optimallocationof pages 17-23) Even though these applications focus on geometric optimization rather than algebraic symmetry, the underlying idea—namely the robustness of diamond–shaped invariances under transformation—is intimately connected to the theorem. Similarly, in the realm of computer graphics and cryptographic visual secret sharing, the diamond theorem provides the structural foundation for generating correlation patterns. In such schemes, 4×4 diamond patterns are sequentially applied to non-overlapping blocks of an image to ensure both secure partitioning and reconstruction of the original visual information. (harish2016newvisualsecret pages 1-2) These diverse applications underscore the theorem’s versatility; its central theme of a combinatorial invariant under a massive symmetry group serves as a unifying idea that transcends disciplinary boundaries.
The explicit description of the permutation group G and the classification of the 840 images into 35 equivalence classes have also motivated algorithmic approaches for pattern generation and classification. For instance, when one wishes to generate all possible G–images of D, it is computationally efficient to recognize that these images naturally fall into 35 distinct classes corresponding to the 35 lines in PG(3,2). Such insights reduce the complexity of computational searches and enable the practical implementation of algorithms in computer graphics, pattern recognition, and combinatorial design. (coqart1978computergraphicsgrid pages 3-3) Moreover, the delta transform method has been implemented in algebraic software packages to construct large rings of symmetric patterns—a development that has implications for both theoretical investigations and real-world problem solving in areas such as coding theory and error–correction. The connection to matrix rings over GF(4) is particularly promising, as it provides an algebraic framework for dealing with vast families of symmetric objects in a systematic manner.
It is instructive to compare the Cullinane diamond theorem with other well-known geometric and combinatorial results. In contrast to classical theorems that rely solely on continuous symmetries or Euclidean transformations, the diamond theorem exploits the combinatorial rigidity of discrete structures. Its reliance on finite fields and projective spaces distinguishes it from many traditional results in geometry. Moreover, while other results in tiling theory or Latin square theory are often limited to ad hoc proofs for specific cases, the Cullinane diamond theorem offers a unifying algebraic–geometric framework that explains not only why symmetric patterns occur but also how they are structured in an entirely discrete setting. This synthesis of group theory, finite geometry, and combinatorial design represents an advance over previous approaches that tended to treat these areas in isolation. (cullinane2013thediamondtheorem pages 1-5, cullinaneUnknownyearlatinsquaregeometry pages 1-6)
The origins of the Cullinane diamond theorem can be traced back to investigations into the symmetry properties of classical tile patterns, including those found in quilts and combinatorial designs. Earlier research, such as that on the delta transforms of the Klein group table, hinted at the possibility that simple tiling arrangements might possess highly non–trivial symmetry properties. Over time, these insights matured into the full–fledged theorem attributed to Steven H. Cullinane, which formalized the connection between a 4×4 diamond figure and the affine group over GF(2). The subsequent discovery of the correspondence between the 840 images and the 35 lines in PG(3,2) further entrenched the theorem’s role as a bridge between discrete combinatorial designs and classical finite projective geometry. In recent years, further work on Latin square geometry and visual secret sharing has expanded the theorem’s impact well beyond its original context, demonstrating that the ideas encapsulated in the diamond theorem are not only mathematically deep but also broadly applicable. (cullinane2013thediamondtheorem pages 1-5, cullinaneUnknownyearlatinsquaregeometry pages 1-6)
The implications of the Cullinane diamond theorem are manifold. On the theoretical side, the theorem points to a rich interplay between discrete geometry, group theory, and algebra that should be explored in greater depth. One promising direction is the extension of the theorem to higher–order arrays and to patterns with more than two colors. Such generalizations would likely lead to the discovery of new equivalence classes and perhaps even connect with higher–dimensional projective spaces. Another area ripe for exploration is the further algebraic analysis of the ring structures generated by delta transforms. In particular, the isomorphism of infinite families of diamond rings to matrix rings over GF(4) suggests deep algebraic symmetries that have yet to be fully exploited. On the applied side, insights derived from the Cullinane diamond theorem have already found applications in facility location, computer graphics, and cryptographic visual secret sharing; further research may reveal additional uses in coding theory, error–correction, and possibly even in the design of secure communication protocols. (cullinaneUnknownyearlatinsquaregeometry pages 1-6, harish2016newvisualsecret pages 1-2)
In summary, the Cullinane diamond theorem is a landmark result that provides a unified explanation for the surprising symmetry properties observed in the four–diamond figure D. When D is subjected to the full action of the permutation group G (of order 322,560), every image obtained—amounting to a total of 840 distinct patterns—exhibits either ordinary or color–interchange symmetry. These images can be uniquely categorized into 35 equivalence classes, each corresponding to a line in the finite projective space PG(3,2). The representation of these classes by three line diagrams whose binary sum is zero not only reinforces the connection with the additive structure of GF(2) but also ties the entire construction to the elegant framework of finite projective geometry. (cullinane2013thediamondtheorem pages 1-5) Further, by introducing delta transforms on square arrays, the theorem extends into an algebraic domain—revealing that the set of symmetric images forms an ideal in a ring of symmetric patterns, with striking isomorphisms to matrix rings over GF(4). This algebraic perspective has fostered connections to Latin square geometry, where the orthogonality of Latin squares is naturally interpreted in terms of skew lines in PG(3,2), and to the combinatorial structures underlying the Miracle Octad Generator and Mathieu groups. (cullinane2013thediamondtheorem pages 1-5, kellyUnknownyearmathieugroupsthe pages 1-1) Ultimately, the Cullinane diamond theorem stands as a testament to the power of combining combinatorial, geometric, and algebraic methods to reveal and classify deep symmetry properties in discrete structures. Its continued influence across diverse fields—from pure mathematics to applied design and cryptographic systems—underscores both its elegance and its far–reaching impact. (harish2016newvisualsecret pages 1-2)
This comprehensive investigation thus not only answers the question “What is the Cullinane diamond theorem?” but also situates it firmly within a larger mathematical landscape that continues to develop and inspire new research.
(cullinane2013thediamondtheorem pages 1-5): SH Cullinane. The diamond theorem. arXiv: Group Theory, Aug 2013. URL: https://doi.org/10.48550/arxiv.1308.1075, doi:10.48550/arxiv.1308.1075. This article has 2 citations.
(cullinaneUnknownyearlatinsquaregeometry pages 1-6): SH Cullinane. Latin-square geometry. Unknown journal, Unknown year.
(giannikos1993optimallocationof pages 17-23): I Giannikos. Optimal location of single and multiple obnoxious facilities: algorithms for the maximin criterion under different norms. Unknown journal, 1993.
[ Correction by Cullinane on June 11, 2025. The Giannikos reference cites a different Cullinane. It is irrelevant and should be ignored. ]
(coqart1978computergraphicsgrid pages 3-3): Roger Coqart. Computer graphics: grid structures. Leonardo, 11:118-119, Jan 1978. URL: https://doi.org/10.2307/1574008, doi:10.2307/1574008. This article has 3 citations and is from a highest quality peer-reviewed journal.
(harish2016newvisualsecret pages 1-2): V. Harish, N. Rajesh Kumar, and N. R. Raajan. New visual secret sharing scheme for gray-level images using diamond theorem correlation pattern structure. 2016 International Conference on Circuit, Power and Computing Technologies (ICCPCT), pages 1-5, Mar 2016. URL: https://doi.org/10.1109/iccpct.2016.7530155, doi:10.1109/iccpct.2016.7530155. This article has 1 citations.
(kellyUnknownyearmathieugroupsthe pages 1-1): S Kelly. Mathieu groups, the golay code and curtis' miracle octad generator. Unknown journal, Unknown year.
" I divide mathematics into discrete and continuous
(prickles and goo, as Alan Watts put it) . . . ."
— Peter J. Cameron on 8 December 2024
* See a YouTube rendition (dated Oct. 28, 2013).
| Number | Space |
| Arithmetic | Geometry |
| Discrete | Continuous |
Related literature —
From a "Finite Fields in 1956" post —
The Nutshell:
Related Narrative:
The Ballad of Goo Ballou —
the Sequel to . . .
“Let me count the ways” is an appropriate request
for students of the discrete , as opposed to the
continuous , which instead requires measurement .
Related academic material —
Raymond Cattell on crystallized vs. fluid intelligence.
For a more literary approach, see Crystal and Dragon
and For Trevanian.
This post was inspired in part by
the American Sequel Society and . . .
The two books pictured above are From Discrete to Continuous ,
by Katherine Neal, and Geometrical Landscapes , by Amir Alexander.
Note: There is no Galois (i.e., finite) field with six elements, but
the theory of finite fields underlies applications of six-set geometry.
This post’s title is from the tags of the previous post —
The title’s “shift” is in the combined concepts of …
Space and Number
From Finite Jest (May 27, 2012):
The books pictured above are From Discrete to Continuous ,
by Katherine Neal, and Geometrical Landscapes , by Amir Alexander.
For some details of the shift, see a Log24 search for Boole vs. Galois.
From a post found in that search —
“Benedict Cumberbatch Says
a Journey From Fact to Faith
Is at the Heart of Doctor Strange“
— io9 , July 29, 2016
” ‘This man comes from a binary universe
where it’s all about logic,’ the actor told us
at San Diego Comic-Con . . . .
‘And there’s a lot of humor in the collision
between Easter [ sic ] mysticism and
Western scientific, sort of logical binary.’ “
[Typo now corrected, except in a comment.]
The ABC of things —
The ABC of words —
A nutshell —
Book lessons —
Paradigms of Geometry:
Continuous and Discrete
The discovery of the incommensurability of a square’s
side with its diagonal contrasted a well-known discrete
length (the side) with a new continuous length (the diagonal).
The figures below illustrate a shift in the other direction.
The essential structure of the continuous configuration at
left is embodied in the discrete unit cells of the square at right.
See Desargues via Galois (August 6, 2013).
* For related remarks, see posts of May 26-28, 2012.
Jamie James in The Music of the Spheres
(Springer paperback, 1995), page 28—
Pythagoras constructed a table of opposites
from which he was able to derive every concept
needed for a philosophy of the phenomenal world.
As reconstructed by Aristotle in his Metaphysics,
the table contains ten dualities….
|
Limited |
Unlimited |
Of these dualities, the first is the most important;
all the others may be seen as different aspects
of this fundamental dichotomy.
For further information, search on peiron + apeiron or
consult, say, Ancient Greek Philosophy , by Vijay Tankha.
The limited-unlimited contrast is not unrelated to the
contrasts between
The books pictured above are From Discrete to Continuous ,
by Katherine Neal, and Geometrical Landscapes , by Amir Alexander.
Commentary—
“Harriot has given no indication of how to resolve
such problems, but he has pasted in in English,
at the bottom of his page, these three enigmatic
lines:
‘Much ado about nothing.
Great warres and no blowes.
Who is the foole now?’
Harriot’s sardonic vein of humour, and the subtlety of
his logical reasoning still have to receive their full due.”
— “Minimum and Maximum, Finite and Infinite:
Bruno and the Northumberland Circle,” by Hilary Gatti,
Journal of the Warburg and Courtauld Institutes ,
Vol. 48 (1985), pp. 144-163
See also Finite Geometry and Physical Space.
Related material from MacTutor—
The paper by J. W. Shirley, Binary numeration before Leibniz, Amer. J. Physics 19 (8) (1951), 452-454, contains an interesting look at some mathematics which appears in the hand written papers of Thomas Harriot [1560-1621]. Using the photographs of the two original Harriot manuscript pages reproduced in Shirley’s paper, we explain how Harriot was doing arithmetic with binary numbers. Leibniz [1646-1716] is credited with the invention [1679-1703] of binary arithmetic, that is arithmetic using base 2. Laplace wrote:-
However, Leibniz was certainly not the first person to think of doing arithmetic using numbers to base 2. Many years earlier Harriot had experimented with the idea of different number bases…. |
For a discussion of Harriot on the discrete-vs.-continuous question,
see Katherine Neal, From Discrete to Continuous: The Broadening
of Number Concepts in Early Modern England (Springer, 2002),
pages 69-71.
“The key is the cocktail that begins the proceedings.”
– Brian Harley, Mate in Two Moves
See also yesterday's Endgame , as well as Play and Interplay
from April 28… and, as a key, the following passage from
an earlier April 28 post—
| Euclidean geometry has long been applied to physics; Galois geometry has not. The cited webpage describes the interplay of both sorts of geometry— Euclidean and Galois, continuous and discrete— within physical space— if not within the space of physics . |
A Log24 post, "Bridal Birthday," one year ago today linked to
"The Discrete and the Continuous," a brief essay by David Deutsch.
From that essay—
"The idea of quantization—
the discreteness of physical quantities—
turned out to be immensely fruitful."
Deutsch's "idea of quantization" also appears in
the April 12 Log24 post Mythopoetic—
|
"Is Space Digital?" — Cover story, Scientific American "The idea that space may be digital — Physicist Sabine Hossenfelder "A quantization of space/time — Peter Woit in a comment |
It seems some clarification is in order.
Hossenfelder's "The idea that space may be digital"
and Woit's "a quantization of space/time" may not
refer to the same thing.
Scientific American on the concept of digital space—
"Space may not be smooth and continuous.
Instead it may be digital, composed of tiny bits."
Wikipedia on the concept of quantization—
Causal sets, loop quantum gravity, string theory,
and black hole thermodynamics all predict
a quantized spacetime….
For a purely mathematical approach to the
continuous-vs.-discrete issue, see
Finite Geometry and Physical Space.
The physics there is somewhat tongue-in-cheek,
but the geometry is serious.The issue there is not
continuous-vs.-discrete physics , but rather
Euclidean-vs.-Galois geometry .
Both sorts of geometry are of course valid.
Euclidean geometry has long been applied to
physics; Galois geometry has not. The cited
webpage describes the interplay of both sorts
of geometry— Euclidean and Galois, continuous
and discrete— within physical space— if not
within the space of physics.
From the current Wikipedia article "Symmetry (physics)"—
"In physics, symmetry includes all features of a physical system that exhibit the property of symmetry—that is, under certain transformations, aspects of these systems are 'unchanged', according to a particular observation. A symmetry of a physical system is a physical or mathematical feature of the system (observed or intrinsic) that is 'preserved' under some change.
A family of particular transformations may be continuous (such as rotation of a circle) or discrete (e.g., reflection of a bilaterally symmetric figure, or rotation of a regular polygon). Continuous and discrete transformations give rise to corresponding types of symmetries. Continuous symmetries can be described by Lie groups while discrete symmetries are described by finite groups (see Symmetry group)."….
"A discrete symmetry is a symmetry that describes non-continuous changes in a system. For example, a square possesses discrete rotational symmetry, as only rotations by multiples of right angles will preserve the square's original appearance."
Note the confusion here between continuous (or discontinuous) transformations and "continuous" (or "discontinuous," i.e. "discrete") groups .
This confusion may impede efforts to think clearly about some pure mathematics related to current physics— in particular, about the geometry of spaces made up of individual units ("points") that are not joined together in a continuous manifold.
For an attempt to forestall such confusion, see Noncontinuous Groups.
For related material, see Erlanger and Galois as well as the opening paragraphs of Diamond Theory—
Symmetry is often described as invariance under a group of transformations. An unspoken assumption about symmetry in Euclidean 3-space is that the transformations involved are continuous.
Diamond theory rejects this assumption, and in so doing reveals that Euclidean symmetry may itself be invariant under rather interesting groups of non-continuous (and a-symmetric) transformations. (These might be called noncontinuous groups, as opposed to so-called discontinuous (or discrete ) symmetry groups. See Weyl's Symmetry .)
For example, the affine group A on the 4-space over the 2-element field has a natural noncontinuous and asymmetric but symmetry-preserving action on the elements of a 4×4 array. (Details)
(Version first archived on March 27, 2002)
Update of Sunday, February 19, 2012—
The abuse of language by the anonymous authors
of the above Wikipedia article occurs also in more
reputable sources. For instance—
Some transformations referred to by Brading and Castellani
and their editees as "discrete symmetries" are, in fact, as
linear transformations of continuous spaces, themselves
continuous transformations.
This unfortunate abuse of language is at least made explicit
in a 2003 text, Mathematical Perspectives on Theoretical
Physics (Nirmala Prakash, Imperial College Press)—
"… associated[*] with any given symmetry there always exists
a continuous or a discrete group of transformations….
A symmetry whose associated group is continuous (discrete)
is called a continuous (discrete ) symmetry ." — Pp. 235, 236
[* Associated how?]
Peter J. Cameron yesterday on Galois—
"He was killed in a duel at the age of 20…. His work languished for another 14 years until Liouville published it in his Journal; soon it was recognised as the foundation stone of modern algebra, a position it has never lost."
Here Cameron is discussing Galois theory, a part of algebra. Galois is known also as the founder* of group theory, a more general subject.
Group theory is an essential part of modern geometry as well as of modern algebra—
"In der Galois'schen Theorie, wie hier, concentrirt sich das Interesse auf Gruppen von Änderungen. Die Objecte, auf welche sich die Änderungen beziehen, sind allerdings verschieden; man hat es dort mit einer endlichen Zahl discreter Elemente, hier mit der unendlichen Zahl von Elementen einer stetigen Mannigfaltigkeit zu thun."
— Felix Christian Klein, Erlanger Programm , 1872
("In the Galois theory, as in ours, the interest centres on groups of transformations. The objects to which the transformations are applied are indeed different; there we have to do with a finite number of discrete elements, here with the infinite number of elements in a continuous manifoldness." (Translated by M.W. Haskell, published in Bull. New York Math. Soc. 2, (1892-1893), 215-249))
Related material from Hermann Weyl, Symmetry , Princeton University Press, 1952 (paperback reprint of 1982, pp. 143-144)—
"A field is perhaps the simplest algebraic structure we can invent. Its elements are numbers…. Space is another example of an entity endowed with a structure. Here the elements are points…. What we learn from our whole discussion and what has indeed become a guiding principle in modern mathematics is this lesson: Whenever you have to do with a structure-endowed entity Σ try to determine is group of automorphisms , the group of those element-wise transformations which leave all structural relations undisturbed. You can expect to gain a deep insight into the constitution of Σ in this way."
For a simple example of a group acting on a field (of 8 elements) that is also a space (of 8 points), see Generating the Octad Generator and Knight Moves.
* Joseph J. Rotman, An Introduction to the Theory of Groups , 4th ed., Springer, 1994, page 2
Catherine Elizabeth "Kate" Middleton, born 9 January 1982,
will marry Prince William of Wales on April 29th, 2011.
This suggests, by a very illogical and roundabout process
of verbal association, a search in this journal.
A quote from that search—
“‘Memory is non-narrative and non-linear.’
— Maya Lin in The Harvard Crimson , Friday, Dec. 2, 2005
A non-narrative image from the same
general time span as the bride's birthday—
For some context, see Stevens + "The Rock" + "point A".
A post in that search, April 4th's Rock Notes, links to an essay
on physics and philosophy, "The Discrete and the Continuous," by David Deutsch.
See also the article on Deutsch, "Dream Machine," in the current New Yorker
(May 2, 2011), and the article's author, "Rivka Galchen," in this journal.
Galchen writes very well. For example —
| Galchen on quantum theory—
"Our intuition, going back forever, is that to move, say, a rock, one has to touch that rock, or touch a stick that touches the rock, or give an order that travels via vibrations through the air to the ear of a man with a stick that can then push the rock—or some such sequence. This intuition, more generally, is that things can only directly affect other things that are right next to them. If A affects B without being right next to it, then the effect in question must be in direct—the effect in question must be something that gets transmitted by means of a chain of events in which each event brings about the next one directly, in a manner that smoothly spans the distance from A to B. Every time we think we can come up with an exception to this intuition—say, flipping a switch that turns on city street lights (but then we realize that this happens through wires) or listening to a BBC radio broadcast (but then we realize that radio waves propagate through the air)—it turns out that we have not, in fact, thought of an exception. Not, that is, in our everyday experience of the world. We term this intuition 'locality.' Quantum mechanics has upended many an intuition, but none deeper than this one." |
An Ordinary Evening in Tennessee
"The rock is the habitation of the whole,
Its strength and measure, that which is near, point A
In a perspective that begins again
At B….." — Wallace Stevens
Related material: The Discrete and the Continuous
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