# Appendix C — Theory of Discrete Manifolds — Part I.—Definitions and Fundamental Theorems

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## Part I.—Definitions and Fundamental Theorems

Section I. In elaborating the theory of Discrete Manifolds, we may, for the sake of simplicity, suppose that all the properties of a given manifold depend exclusively on the Facts of Nextness which obtain in regard to the elements of which the manifold is composed. One feature which differentiates Discrete from Continuous Manifolds is that the former are actually composed of their elements, whereas Continuous Manifolds are, as is well known, something more than mere aggregates of their elements. An element of a Continuous Manifold is a boundary between two adjacent regions of the manifold; and hence, as Professor Clifford points out in his lecture on the "Postulates of the Science of Space," two adjacent regions of a Continuous Manifold have the same boundary, whereas two adjacent regions of a Discrete Manifold have two different boundaries—namely, the terminal elements of the two adjacent regions. As Professor Clifford remarks, this particular distinction between Continuous and Discrete Manifolds was already known to antiquity.

Section 2. The assumption made in Section 1 that the properties of any particular Discrete Manifold to be treated shall depend exclusively on the Facts of Nextness which obtain in regard to its elements, is not only a convenient one for the purpose of limiting an investigation which would otherwise cover too large an area, but it may also transpire that it has an important bearing on ultimate mechanical page 38 theories of the physical universe, if the prejudice [unclear: ent] tamed by many physicists and philosophers against [unclear: admitti] the possibility of "action at a distance" should ever [unclear: tu] out to be in harmony with fact. For it is conceivable [unclear: th] an ultimate mechanical explanation of the universe [unclear: m] involve the assumption that it is a Discrete Manifold [unclear: exi] ing in a Time which is itself Discrete, or composed of [unclear: i] divisible elements of duration; and further, that an [unclear: elema] of the physical universe can only influence its immedist neighbour or neighbours in the next instant of Time, [unclear: an] requires a plurality of such instants in order to [unclear: influency] even in the slightest degree, more distant elements. [unclear: W] might thus get a mechanical explanation of the [unclear: uni] founded on the denial both of continuity and of "action [unclear: a] a distance." Or, as has been suggested by Dr. [unclear: Theodo] R. Noyes, elements of the discrete manifold constituting [unclear: th] physical universe may periodically appear and [unclear: disaper] (come into existence and cease to exist) at differing units [unclear: o] Time, thus giving rise to the phenomenon of wave-motio because altering the "facts of nextness" in successive [unclear: sma] regions of space.

Section 3. "Nextness" is, of course, assumed to be mutual or reciprocal fact. If A is "next" to B, B, [unclear: o] course, is "next" to A. Under such circumstances, I [unclear: sh] call A a neighbour of B, and B a neighbour of A. [unclear: Thi] nomenclature is essential in order to avoid an [unclear: awkwa] periphrasis.

Section 4. We shall further assume that all the Facts of Nextness obtaining with regard to the elements of a Discrete Manifold consist exclusively of statements respecting the number and identity of the various neighbour which each element of the manifold possesses. This assumption at once differentiates Discrete Manifolds into two greaf classes—(a) those in which each element has the same number of neighbours, and (6) those in which each element has not the same number of neighbours. The former we shall call Homodromic Manifolds, and the latter Hetero-dromic Manifolds. The ground of this terminology lies [unclear: is] the fact that the number of neighbours which a given element page 39 of a manifold possesses determines the total choice" in respect to initial direction of setting out which a traveller would have in passing from a given element to other elements of the manifold. It is thus analogous to the total quantity of turning or angular magnitude which exists at any point of a Continuous Manifold, and it is therefore appropriate, where this total choice is a constant quantity, to use the term Homodromic, and to use the term Hetero-dromic where the said total quantity of choice varies from one element to another of the manifold.

Section 5. The simplest of Discrete Manifolds is, of course, one which contains only two elements—say A and B. There can be no diversity in regard to the "Facts of Next-ness" in this manifold. Each element must have one, and only one, neighbour.

Section 6. The next simplest case is that of a manifold having three elements. Here there are two possibilities in regard to the Facts of Nextness. It may be that one of the three elements has each of the two others as a neighbour, and that these two others are not next to each other, in which case we have a Heterodromic Manifold. Or, on the other hand, it may be that each element has both the others as neighbours, in which case we have a Homodromic Manifold. A Homodromic Manifold of three elements would, of course, be one where a traveller who never stopped or re-traced his steps would of necessity make a cyclical journey by a unique route (the identity of a "route" being supposed independent of the order in Time in which it is traversed). Every Manifold of which this is true must consist of a finite number of elements, each having two, and only two, neighbours; and we shall call such a Manifold a Cycle.

Section 7. Four is the smallest number of elements which a Discrete Manifold can possess in order that any one element may have more than two neighbours. It is dear that, in a Discrete Manifold of four elements, one or more of those elements may have three neighbours. If each of the four elements has three neighbours, we obtain a Homodromic Discrete Manifold, which is analogous to the page 40 face aggregate of a tetrahedron. On the other hand, [unclear: i] for example, only one element of the four has three [unclear: nei] bours, while each of those other three has only the [unclear: sa] first element as neighbour, we obtain a Heterodromic [unclear: Ma] fold analogous to the chemical formula of ammonia.

Section 8. Leaving now for a moment the subject Finite Manifolds with a very small number of element-subject which has been sufficiently treated in order to [unclear: shoo] what is really meant by the terminology here adopted. We observe that Homodromic Manifolds may be [unclear: classifi] according to the number of neighbours which each [unclear: eleme] has, and, further, that an Infinite Manifold (that is, a [unclear: ma] fold containing an infinite number of elements) must [unclear: b] such that each element has in general at least two neighbour.

Section 9. Passing to the consideration of Infinite [unclear: Ma] folds, the simplest case is obviously that in which [unclear: ead] element has two neighbours. Such a manifold I call [unclear: a] Infinite Chain. One of its properties is clearly that traveller, in starting from any particular element and [unclear: setting] out in either of the two directions open to him, could [unclear: neve] again (without retracing his steps) arrive at the element from which he set out, for it will be found that if the [unclear: chai] crossed itself anywhere there would at the intersection [unclear: b] an element having more than two neighbours, An [unclear: obvios] complication on the "Infinite Chain' is the case of manifold in which a certain element, to be called the [unclear: Initi] Element, has only one neighbour, whereas every other [unclear: elements] has two neighbours. An example of such a manifold [unclear: i] furnished by the series of positive integers where the number has only one neighbour (viz., the number 2), whereas each other number has two neighbours. Such a manifold [unclear: w] might call an Infinite Ray.

Section 10. The next simplest instance of an Infinite Homodromic Discrete Manifold is that of a manifold when each element has three and only three neighbours. It will be found on investigation that this manifold may share the property of the Infinite Chain of being so constituted that a traveller setting out from a particular element, and [unclear: at] no time retracing his steps, could never return to that page 41 element. When it possesses this property I shall call it a "Tree."

Section 11. The smallest number of neighbours which the element of an Infinite Homodromic Discrete Manifold can have, in order that the above-named property shall not be inevitable, is three. Where the elements of a manifold each have three and only three neighbours, there, even although the manifold may be infinite, it does not follow that a traveller setting out from a particular element (and never reversing any step of his journey—this is, of course, assumed in all that precedes) cannot return to his starting point. It may be so or it may not be so. An Infinite Homodromic Manifold in which there is this impossibility of returning to the starting point, I shall call, as before stated, a "Tree," whereas an Infinite Homodromic Manifold in which the cyclical return spoken of is possible, I shall call a Net. It is obvious that "Trees," while they cannot be so constituted that an element has less than three neighbours, may be so constituted that each element has four or any larger number of neighbours. We might classify such manifolds as triply branched, quadruply branched, &c., Trees.

Section 12. The case of an Infinite Homodromic Discrete Manifold in which each element has four and only four neighbours, and in which cyclical return is possible, is one of the simplest cases of a Discrete Manifold which brings us en rapport, so to speak, with the doubly extended Continuous Manifolds that are familiar to us all as surfaces or "two-way spreads." I think it will be found that the above-mentioned simplest Discrete Manifold of this type, viz., one where each element has four and only four neighbours, and where cyclical return of the type A, B, C, D, A is every-where possible is, when taken on a large scale, approximately identical in some of its properties with the Euclidian plane, though differing from the latter in being æolotropic (of "crystalline" structure, so to speak) instead of isotropic, and such that the ratio of the contour which most nearly satisfies the definition-property of the circle, to a diameter, is 4 instead of π. It would appear, however, that there page 42 is another Infinite Homodromic Discrete Manifold which all has somewhat similar characteristics when taken on a large scale, viz., the manifold where each element has six and only six neighbours, and where cyclical return of a still [unclear: mon] immediate type (viz., A, B, C, A) is everywhere possible.

Section 13. It will be found that in the case of "Trees" there is no meaning in predicating a number of "[unclear: dime] sions" : but in the case of "Nets" the number of dimension will be found to depend on the relation between the differed routes of shortest cyclical return which can be traversal when setting out from an element. The comparison of [unclear: a] number of counters arranged in contact with each other [unclear: on] a table so that each counter has four neighbours, with [unclear: a] heap of cannon balls so arranged that each ball has [unclear: six] neighbours, will both suggest and illustrate this fact.

Section 14. It will be seen from the above that one of the purposes to be kept in view in elaborating the theory of Discrete Manifolds, is to affiliate the various types [unclear: of] such manifolds to the analogous Continuous Manifolds which are familiar to us in geometry and in the theory of Continuous Extension generally. To work out this relation or analogy in detail will be the object of a future paper. The purpose of the present one is merely to indicate the line [unclear: of] investigation which will have to be pursued in order to attain this end, and to show that in the case of Discrete Manifolds the theory of the subject can be built up by a strictly synthetic process, whereas, in the case of Continuous Manifolds, owing to the impossibility of deriving the properties of the manifold from Facts of Nextness obtaining in regard to its elements (since, in a Continuous Manifold, no two elements can be "next" to each other), a combination of analytical with synthetic treatment is indispensable.

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