Negative Dimensionality

by Tom Veatch
September 18, 2015

In 1978 with NSF support I spent a summer studying David Hilbert's axiomatic approach to geometry under Edmund Deaton at San Diego State University. Thanks to both of them, and to you American taxpayers for paying the taxes that fund the NSF!

Hilbert axiomatized geometry as follows:

  • I Axioms of Incidence
    • I.1 For every two points A, B there exists a line a that contains each of the points A, B,
    • I.2 For every two points A, B there exists no more than one line that contains each of the points A, B.
    • I.3 There exist at least two points on a line. There exist at least three points that do not lie on a line.
    • I.4 For any three points A, B, C that do not lie on the same line there exists a plane α that contains each of the points A, B, C. For every plane there exists a point which it contains.
    • I.5 For any three points A, B, C that do not lie on one and the same line there exists no more than one plane that contains each of the three points A, B, C.
    • I.6 If two points A, B of a line a that lie in a plane α then every point of a lies in the plane α.
    • I.7 If two planes α, β have a point A in common then they have at least one more point B in common.
    • I.8 There exist at least four points which do not lie in a plane.
  • II Axioms of Order
    • II.1 If a point B lies between a point A and a point C then the points A, B, C are three distinct points of a line and B then also lies between C and A.
    • II.2 For two points A and C, there always exists at least one point B on the line AC such that C lies between A and B.
    • II.3 Of any three points on a line there exists no more than one that lies between the other two.
    • II.4 Let A, B, C be three points that do not lie on a line and let a be a line in the plane ABC which does not meet any of A, B, C. If the line a passes through a point of the segment A B, it also passes through a point of the segment AC or through a point of the segment BC.
  • III Axioms of Congruence
    • III.1 If A, B are two points on a line a, and A' is a point on the same or on another line a', then it is always possible to find a point B' on a given side of the line a' through A' such that the segment AB is congruent or equal to the segment A'B'. I.e., AB ≡ A'B'.
    • III.2 If a segment A'B' and a segment A''B'' are congruent to the same segment AB, then the segment A'B' is also congruent to the segment A''B'', or briefly if two segments are congruent to a third one, they are congruent to each other.
    • III.3 On the line a let AB and BC be two segments which except for B have no point in common. Furthermore on the same or on another line a' let A'B' and B'C' be two segments which except for B' also have no point in common. In that case, if AB ≡ A'B and BC ≡ B'C', then AC ≡ A'C'.
    • III.4 Every angle in a given plane can be constructed on a given side of a given ray in a uniquely determined way. Formally: Let ∠(h,k) be an angle in the plane α and a' a line in a plane &alpha' and let a definite side of a' in α' be given. Let h' be a ray on the line a' that emanates from the point O'. Then there exists in the plane α' one and only one ray k' such that the angle ∠(h,k) is congruent or equal to the angle ∠(h'k') and at the same time all interior points of the angle ∠(h'k' lie on the given side of a'.
      Every angle is congruent to itself.
    • III.5 If for two triangles (vertices not on the same line) ABC and A'B'C', the congruences AB ≡ A'B', AC ≡ A'C', ∠BAC ≡ ∠B'A'C' hold, then the congruence ∠ABC ≡ ∠A'B'C' is also satisfied.
  • IV Axiom of Parallels
    • IV (Euclid's) Let a be any line and A a point not on it. Then there is at most one line in the plane, determined by a and A, that passes through A and does not intersect a.
  • V Axioms of Continuity
    • V.1 (Measure, or Archimedes') If AB and CD are any segments than there exists a number n such that n segments CD constructed contiguously from A, along the ray from A through B, will pass beyond the point B.
    • V.2 (Line Completeness) An extension of a set of points on a line with its order and congruence relation that would preserve the relations existing among the original elements as well as the fundamental properties of line order and congruence that follows from Axioms 1-3 and 5.1, is impossible.

Thoughtful as Hilbert is, it seems simpler would be better. For example:

  • 1.1 and 1.2 could be unified by replacing "a line" and "no more than one line" with "one and only one line", for example, "one and only one" is used in 3.4.
  • Similarly 1.4 and 1.5 could be unified via "one and only one plane".
  • Indeed, reduce "one and only one" to "one"; a small matter of semantics.
The logical meaning of "one" is profound enough, but the big deal here, that draws my attention, is the great and most profound redundancy and uneconomy of not just Hilbert's but our whole geometrical way of thinking, which Hilbert expressed. There is a higher level sameness of repeated structure in this work. Do you see that the construction of higher-dimensional spaces from lower-dimensional spaces follows the same path, going from points to line, lines to plane, planes to space (and you can imagine continuing into higher dimensions indefinitely)? Or in reverse, that these vast inclusion relationships increase the numbers so infinitely as we go the other way around, since one space contains such a multitude of planes, one plane such a multitude of lines, one line such a multitude of points. These statements are so similar, there's so much redundancy going up and down these dimension-counting dimensions that I feel drawn to simplify. It could lead to deeper understanding, or at least to fun. There is some fun here! Allow me therefore to try a more minimalist axiomatization, indeed one which could extend to higher numbers than three, or indeed to lower numbers than zero:
  • Definitions:
    • First, define sets, set equality, unions, intersections. You can do it. Also let "distinct" mean "having intersection not equal to union".
    • Second, define "Geometry" as a "set allowing extension of membership subject to Geometric Axioms".
    • Third, define "Geometric Combination" as between two Geometries and with two properties, as follows:
      • For two distinct Geometries of the same dimension,
      • with non-empty intersection:
      • they Geometrically Combine such that there exist unique Geometries of adjacent dimension comprising their
        • Geometric Intersection, of lower dimension
        • Geometric Projection, of higher dimension
  • Geometric Axioms:
    • G.1 A Geometry of no dimension (i.e., where D = 0) exists (call it a "point").
    • G.2 If one D dimensional Geometry exists, then another, distinct one also exists.
    • G.3 If two D-dimensional Geometries are distinct and intersect each other, then they Geometrically Combine.

G.2 captures I.3, I.8, while G.3 captures 1.1, 1.2, 1.4, 1.5, 1.6, 1.7. For two distinct-but-intersecting D-dimensional geometries their intersection is a unique D-1 dimensional space, and they project a unique D+1 dimensional space. Thus lines(D=1) intersect if at all in a unique point(D=0); planes(D=2) intersect if at all in a unique line(D=1); presumably we can say "etc., etc. for higher dimensions". Intersecting geometries have a full sub-geometry in their intersection (1.7) so that the intersection of planes is a line, not a point (etc., etc.)

Professor Harry Goheen pointed out in his Forward to Hilbert's Foundations of Geometry, Hilbert showed not only a "model for the axioms of geometry" but also proved that "any [other] model is isomorphic to [his]. It may therefore be left as an exercise for the reader to add to the above Geometric Axioms what is isomorphic to Hilbert's other Axioms of Order, Congruence, Parallels and Continuity.

This axiomatization, like Hilbert's, was first drafted constructively, or procedurally, in a form similar to computer code whereby one starts at zero, and moves incrementally up through the dimensions. But I thought, let's try to build it functionally, so that the relations between higher and lower dimensions can be specified, and a whole infinite structure simultaneously given its laws, so that a person or computer thinking about it could travel not just upward and not just up to three, but farther, and also downward, and not just down to zero, but farther.

Projection for example could be defined using a Geometry plus a Point outside it, as Hilbert did, but that approach might not work with negative dimensions, and if we want to consider dimensionality in the abstract, then getting away from particular dimensionalities like D=0 Points, would seem a step in the right direction. Perhaps G.1 could be done away with, considering this recurrence is more local and uses only adjacent-dimension Geometries.

Occam's Razor, the force of economy of thought, has led me to propose G.1-3 as replacements for I.1-8, but also leads to a surprising outcome.

This axiomatization offers the possibility, indeed requires, for it to work, that there exist Geometries of negative Dimension (which some might consider a priori as a reason to discard the whole, but for the sake of argument, let us proceed, I don't think we have contradicted ourselves yet).

What is the intersection between points? Naively one could say that if they are the same point, the intersection is the union, they are not distinct; and if they are distinct, not the same point, then there is no intersection, or the intersection is empty. But in the current view, a point is just an N-dimensional Geometry where N=0, and (following G.3), Geometries of yet-lower dimension, intersecting yet distinct from each other, somehow Project the next higher-dimensional Geometry. So given two distinct-but-intersecting D=-1 dimension Geometries, they would Geometrically Combine (G.3) meaning they Geometrically Project a D=-1+1=0 Geometry, namely a point. If we take this for granted as obviously true for D=0's (points) projecting D=1's (lines), and for D=1's(lines) projecting D=2's (planes), and D=2's(planes) projecting D=3's(spaces) and imaginably into higher dimensions of time etc. (see footnote 1, below), then why shouldn't we be open to thinking it might be true with D=-1's projecting D=0's? And the same argument goes for intersections, going from D=3 down to D=0. I have forced myself to be open to persuasion on this, sorry for the pun, point! Let us say it can, then.

So then let us comfortably consider that a point indeed does not, cannot, intersect as a point with another distinct point (otherwise they would be the same point rather than distinct points), so far so good, but I note, rather less comfortably, that this might mean that G.3 could not apply to project a line from these two points. See my issue? (The conditions in G.3, for projecting higher dimensions and intersecting in lower dimensions, are the same conditions, so if we want to project lines from points, we have to also meet the conditions and be able to Intersect points to make D=-1-dimensional Geometries).

Perhaps we can imagine, then, a Geometry in which points intersect one another in negative-dimensional sub-Geometries of Dimension D=-1 where there is a D=-1-dimensional overlap between two D=0-dimensional points, and in such a case two points, distinct, but now in a lower dimension intersecting, D=0 dimensional Geometries, could therefore be considered using G.3 to Geometrically Project a D=1-dimensional (line) Geometry. So negative dimensionality rescues G.3, enables us to go up as well as down from D=0 points and thus to project lines from points, and in short we can go ahead and believe that one generalization applies to all the dimensions up and down.

So one might say Reality Intersects Theory only for dimensions 0 to 3, plus or minus 1 (we need D=0-1 so that Projection can apply to something to bring us D=0 Points, and D=3+1 is needed so that Projection of 3D spaces is tolerated too, as G.3 requires).

The mental projection of higher dimensional Geometries than the third dimension, for example, on analogy with a time dimension or with higher-dimensional ordered n-tuples of vector spaces are of course part of learning higher math.

But what it might mean to be negative-dimensional, somehow connecting points outside of space and time, may be a Zen exercise for the intuitive, but perhaps formally no more so than to imagine "point" objects of zero spatial extent. In any case it can be no objection to negative-dimensional Geometries that there is nothing, or less than nothing, there, since there is nothing in a point, either.

One possible analogy for negative-dimensional geometries might be taken from the intersecting of a line(D1) with a plane(D2) in a point(D0): imagine a sort of interpenetrating space with a locally conical set of rays coming from a D=-1 de-point located out there somewhere, sending rays to every actual D=0 point in the universe, and thus rather conically sending rays to all the points around here, for example, and having those rays pick out and define actual points in actual space only upon intersecting with the higher-dimensional space of Reality. The analogy seems imperfect since lines-intersecting-planes-in-points is a D1/D2/D0 relationship while the analogy proposes imagining a D-1/D3/D0 relationship.

This analogy might tempt a physicist to consider whether Gravity, which interconnects all space through some mysterious fabric ungoverned by the normal laws of space and time, is a Geometry of negative dimension. But no, subGeometries are smaller, so I would consider Gravity the other way around, as a phenomenon expressed within the Geometric Intersection of some higher-dimension Geometries.

Another imaginable view on dimensionality is as a sliding window on information dimensions. In the conceptualization of any set of continuous dimensions, one could sort them in one direction or another, and one could start in the middle of a list, for example. Then starting from some arbitrary index within the list, if we pick the next three values, we can envision them as distributed in a 3D space. Well, equally we could go backwards in the list and talk about the values preceding those selected by that arbitrary index. Which dimensions are those, when they precede our arbitrary starting point? In such a system, dimensionality is rather like attentional focus, in that we may say the dimensions we are looking at now correspond to a 3D space, but we might shift out attentional focus to a different set, a re-ordered set, or if an ordering is externally imposed, then a set preceding in the order the ones we were looking at a moment ago, and these would be the negative dimensions of our informational space. Along these lines, the dimensions of a given 3D Euclidean space may be considered as based on viewing as constant the contextual information or preceding dimensions, for example, viewing time or any other analytic precondition as having a contextually-specified value and being thus the preceding or lower dimension (D=-1) in an ordering of the dimensions that places the properly spatial dimensions as dimensions 0, 1, and 2.

In such a view, time is a negative dimension, not the fourth dimension, but it may be stressed that these Geometries, Dimensions, and Orderings are all analytical conveniences taken up in one view or another but not exactly representing the Truth out there, which is a matter more of Physics than this abstract Geometry. So G.1 and G.2 which assert a kind of Reality on which to base G.3, could be internally dispensed with, and supplied by external considerations: the analyst who decides what are the dimensions of interest, how to group them together, which to focus on, etc., may be depended upon to supply the initial Point of G.1 and distinct-yet-intersecting Other Geometries of G.2. The job of abstract geometry's 8 Incidence Axioms are here reduced to a single statement, G.3, and the job of applied mathematicians is to separately put forth objects of their own understanding and interest, to which G.3 then applies, in order to think Geometrically about their problem.


Is it time for some terminology? So the English morpheme "de-" means "away from", shares /d/ with "dimension" and shares "de" with Dr. Deaton my teacher, and while I don't wish to be mean to Hilbert whose shoulders we are standing upon here, "de-" also shares initial /d/ with Dilbert, a sort of anti-Hilbert. So perhaps we might call, for convenience of reference, a D=-1 Geometry a de-point, a D=-2 Geometry a de-line, and a D=-3 and D=-4 Geometries de-planes and de-spaces. Maybe unpoints, unlines, and unplanes will prove more popular, but I'll use de- for now.

Let us clarify the requirements of G.3 here: We would need two distinct D=-1 de-points Geometrically Intersecting each other (in a D=-2 de-line) in order to Geometrically Project a D=0 point. Similarly two distinct D=-2 de-lines Geometrically Intersect in a D=-3 de-plane and Geometrically Project a D=-1 de-point. The recurrence relationship of G.3 can be followed to send our thinking back to de-spaces, etc. ad infinitum, as far as you may like.

Now it is clear that the names have a reversal in them in that de-point vs de-line reverses the relationship of intersection and projection found between point and line: points project lines and are their intersection, while de-lines project de-points and are their intersection. Yes, according to G.3. However! the difference between projection and intersection is primarily that they are opposite in picking out a higher-or-lower dimensionality of the uniquely so-selected adjacent-dimensioned Geometry, but the construction is almost perfectly symmetrical and so it is worth considering whether the polarity we happen to choose first is non-essential.

If we started with de-points instead of points by renumbering dimensions as d = -(D+1), and relabelled "intersect" and "project" each as the other, then would the same properties of Geometry still continue to follow throughout? Is there much difference between the world we think we are actually in, with D=0..3, and the de-world with D=-1..-4? If we were in the other, would it make any difference?

The only difference is the direction of inclusion when we use the word "distinct", when distinct Geometries intersect, just as for example, the only difference between the Fourier transform and its inverse Fourier Transform is a negative sign, or between considering a line with one end in the positive direction and itself the other way around, is that single bit of orientation, even while multiple levels of infinity are packed into any segment thereof. We could swap intersection and union, along with Geometrically Intersect and Project, and have the same world inside out.

So we can in imagination turn the dimensional increase and decrease around the other way and get a de-Geometry which looks rather similar to the Geometry. Now I think I have proven:

Math Is Fun
Q.E.D.
 


 

Footnote 1: We have stretched our minds to think of negative dimensions in which D=0 points might intersect so as to enable G.3 to apply thereby gaining as a theorem the proposition that two points project a line. Let us also briefly consider how two D=3 spaces might intersect to make a plane. Two 3D spaces might conceiveably (a) not intersect at all, or (b) if they did intersect it might take some thought to imagine how. Consider as an example time as a fourth dimension. Then an entire Euclidean 3-D space at one moment in time would have no intersection at all with the same entire Euclidean 3D space at any different moment in time, because nothing in the first is at the time of the second. Thus (a). How then might we construct (b), an intersection of 3D spaces? As a thought experiment imagine something traveling like a photon straight through a 3D universe from one end to the other, but having something like a 360-degree camera looking out in all directions but only perpendicularly to its path. Then every point in the universe would at some time be scanned by our travelling time-indexing camera, and every point that is scanned at the same moment is in a plane with all the other such points. Now this collection of planes at different times can be considered to be a 3D space, because the planes are 2D and the travelling camera's location-and-time is a third dimension projecting planes along its infinite flight path. It just has newer planes in one direction, and older planes in the other direction, but is equally infinite, continuous, and geometrically dimensional as any other 3D space one might imagine. So far so good. Now what is the intersection of this traveling space with our usual Euclidian 3D universe -- taken of course at one time, say, now? Of course the intersection is that plane picked out by our travelling infinitesimal camera at the current moment!

So consistent with G.3, we have two intersecting-but-different 3D spaces, and therefore they must Geometrically Combine, so they Geometrically Intersect in a space of the next lower dimension namely a 2D space or a plane, and indeed we find in this thought experiment that they would do so. Super! G.3 also states that they would Geometrically Project a 4D space, and indeed an entire 3D universe taken with time as a 4th dimension is certainly Projected by our intersecting moment-space and travelling-camera space.

September 29, 2015

Footnote 2: A challenge stands here: can all of geometry be derived from an axiomatic treatment built around the general rule of combining geometries, G.3? It is an open question.
  • Hilbert, Moore, etc., leave "point" undefined, whereas here point, line, plane, space, etc., are defined as geometries of dimension 0, 1, 2, 3, etc., respectively.
  • The zero extent, or infinitesimality, of geometries of lower dimension within geometries of higher dimension is suggested by G.3: if there were some non-zero thickness of point within line, line within plane, etc., then projection of that thickness outward and inward could get us the higher dimension without the intersecting-but-distinct geometry provided to G.3, so G.3 would be unnecessary. Is that a proof? Not yet.
A world of work remains to be done. October 6, 2015
Footnote 3: I asked Dr. Deaton what he thought of this; he asked some mathematician colleagues; they said this is "uninteresting" because there aren't any existing objects they can imagine that can be thought of using de-points, de-lines, etc.

Rebuttal:

A. Since when has that been an objection in mathematics?

B. If you want to be able to think of an application of this to something more concretely imaginable, start like this. Suppose you're interested in measuring stuff. Suppose you can make a lot of different measurements. Well, now you need high-dimensional vectors. So let's consider a vector of, say, 8 (or any number of) real numbers. Select some one in the middle somewhere, say, #4, as our arbitrary reference "zero'th" dimension, and for convenience of imaginary representation, identify dimensions #4, #5, #6 as spatial dimensions that we may familiarly reference as x, y, z, and #7 as a temporal dimension t. These identifications are convenient so as to imagine subsets and geometries within this vector's range of values, but they are not essential, since space and time could be identified with any other subset of 4 out of the 8 dimensions, and subsets and geometries defined and visualized with those indexes instead.

Let c1,...c8 be distinct, constant, real numbers. Let v1,...v8 be real variables (each a name for the set of reals).

Now, what we might in normal cartesian geometrical discussions refer to as the "point" (x,y,z) = (0,0,0) at time 0, is actually, within this scenario an object which may have more or less specification of values in the unlisted dimensions. In my imagination I tend to populate the unseen dimensions with some constants. (As in, a space is a space at a particular time. Or as in, a line along the z dimension is a line at some value of x and some value of y.) But this just makes explicit what kind of context the Geometry under consideration is interpreted relative to, and one could generalize rather than particularize, and we will see what is more convenient as we go. Starting with particularized contexts, then, let's examine a set where the preceding dimensions #0, #1, #2, #3 would be defined as specified with certain constant values so that the fully specified origin in space and time is actually some [c1,c2,c3,c4,0,0,0,0] with constants specified in the other, preceding, hence "negative", dimensions. With those values of the negative dimensions given as constants, we are looking within an infinitesimal subset of our vector space. A completely different origin point is located at [c1,c2,c3,c4+1,0,0,0]. Within one context, now, certain "lines" might be orthogonally constructed in this infinitesimal subset of R^8 by combining [c1,c2,c3,c4,0,0,0,0] and [c1,c2,c3,c4,1,0,0,0], for a simple example, to get [c1,c2,c3,c4,v5,0,0,0] where v5 now varies along what we might call the x axis, or [c1,c2,c3,c4,0,v6,0,0] where y varies, along its own line, etc. (Note that it doesn't much matter if x is first or y or z, it's a line any way you sort the dimensions in our vector. This generalizes.) Similarly projecting up we might combine a line along x with a line along y to get the x-y plane, as the set [c1,c2,c3,c4,v5,v6,0,0]. Intersection operates also as per G.3 where the v5 and v6 values are set to the values specified in the intersecting set so that we get [c1,c2,c3,c4,0,0,0,0] once again, a D=0 point.

Now, we have seen how extra dimensions can hang around as context and not damage our intuitions about projection and intersection operations for more normal geometries, but in this generalized multidimensional geometrical view what would be an example of a de-point, a D=-1 geometry? It is smaller than a point, right? Okay this is where we see that my imagination's tendency to provide constants for the context dimensions was the wrong idea. If a point has variables rather than constants in its context, then setting one of them to a constant would reduce the size of the resulting geometry. So the origin point should be [v1,v2,v3,v4,0,0,0,0] with variables in the context. And some de-points that might combine projectively to generate that D=0 point would include [v1,v2,v3,0,0,0,0,0] and [v1,v2,v3,1,0,0,0,0]. See how they project, since the real variable v4 can be represented as a weighted combination of 0 and 1? Next, well, those particular de-points have no intersection, but de-points [v1,v2,v3,0,0,0,0,0] and [v1,v2,0,v4,0,0,0,0] do intersect at a D=-2 de-line [v1,v2,0,0,0,0,0,0]. Looks like we have something here.

So you mathemticians, you want to tell me G.3 is uninteresting and meaningless? It doesn't establish a foundation for anything? Hmph. These operations projecting up and intersecting down, on any sorted list of real values, are perfectly sensible representations for perfectly well-defined mathematical objects. And to my mind because now you're forced to be explicit about the invisible context dimensions, whether they are populated with constant values, or left as variables, etc., you can have more clarity in functions or operations in numerically describable many-dimensional geometries.

Starting anywhere in the R^N vector. You can intersect down and project up, all you want, and get perfectly well-defined subsets of N-space. This lets you play in high-dimensional spaces of arbitrarily ordered dimensions, using the tools of geometry, consistent terminology, general concepts, and clarity about your context. Dude, clarity helps.

February 24, 2017

 

 

Copyright 2015-2017 Thomas C Veatch