note to all nD-applets makers

[thigle]

1. few things that you people who program these wonderful hyperdimensional snapshots, sequences & even raytraycers are left with, to code 4 dimensions into 2d plane of representation, are, usually, some color-coding and a bit of renaissance perspective.

noticing the illustrations in Geometry & Imagination by Hilbert/Cohen-Voss, the diminishing thickness of lines in perspective is used skilfully to evoke experience of depth (3d).

could someone implement this (pat, quickfur or others skilled in programming...) ? this would let another code-spectrum to be interpenetrate with those currently used, which would allow easier 'mind-plementation' on user's part.

2. (the following is partly spaced out).
the space from which we project is the deeper intent of our projection-making - i.e. we try to acquire a different than usual intuition of n-space by understanding operations and structures within questioned space.
however, that (mind)space which is our major habitual tendency (for our spatial intuition) the cartesian 3-space which is thus almost-hard wired to E3-structuring (& therefore inhibits mostly the acquiring of another in-tuition for space), is the space which is used for almost all the projections I've seen up to this day (perhaps apart from few, like for ex. burbanks', which use a wide FOV projection, however, in flat conditions (zero curvature)). this state of affairs contradicts its own goals, (unique novelty cannot ) and should be corrected.

if you look at the actual visual field in its wholeness, our vision is actually curved the closer to infinity (=farther from frontal focus) it gets (border of vision is horizon defined by eye-holes in our flesh). also the retina is curved surface. open-landscape dwellers tend to mentally curve straight parallel lines, contrary to habitual tendency of urban humans, who have in general tendency to straigten/flatten up slightly curved parallels into straight lines. already leonardo explicitly stated and sketched on curvature of visual space. however, the flat version of perspective prevailed along with flat rationalism and re-wrote experiential, curved space of vision, into rationalized abstraction of E3.
thus, abstraction took value over experience-as-such. such a reduced (non-integral) mode of mind, similar to thinking (not so many ceturies ago) that world is flat, until it was found that it is (at least locally) multiply connected. then Einstein, forcing the cosmological constant into his equations with (maybe unconscious) intention to keep the old flat-mind consesus/habit alive, forced the curvature (=roundness, feminine) out of focus. in the same way, Vision was forced into seeing, even though the seer has been banned from his own seeing by this very act. our souls were hidden to us by ourselves in the night. but Vision - experience-as-such, is naturally multiply-connected, round and ever-returning, not simple, flat and fixed. Turner, Constable, Escher, Flocon&Barre, Termes and recently many others, have shown the many possibilities in which the visual reduction can be released back into its original curved space of silence.

and so i think it would be appropriate to let the old habit go, and form a new one. here is a sequence of perspectival templates, all different from currently used.
a. the first simple? step1 would be to move the projections into simply curved, half-spherical perspective. good exemple is for exemple 180degreeFOV fisheye-lens camera, or exlorations of concave/convex mirrors, glass-spheres... this means unifying the horizon(real line) into a 'circle'(real projective line). expanding into, and embracing from, the shining limit.
b. then step2 would be making it panoramatic - i.e. 2 half-spheres of vision into one whole sphere of vision. front & back of visual space connected as if rotating the viewer 2pi around the vertical (or world around the viewer). two fisheye-views / 2 circles-borders, lines at infinity, touching at single point (left or right). merge the neighbouring halves of 2 infinity-borders, from zenith through point of contact to nadir, and rearrange, similarily to grid transformations of d'Arcy Thompson) into a unity (circle) again.
c. step3 - integrating verticality. we get a 3torus (i think. ?) (or is it a glome?), where each perspective-frame in vertical circular sequence is one complete RP3 (E3+RP2)?. 2 RP3 with their infinity-boundaries. the point of view is from (any?) one point-hole, taken out of RP2. thus an unfolded klein from 2 mobius-strips joined along their boundaries, is left as the patch of fabric, from which this planar representation of whole sphere of vision is woven. this is quaternion-level, as each quaternion on unit sphere corresponds to one viewpoint(or viewing axis, if considered as vector), opposites not identified.
step4 - diagonal yoga: seems to correspond to octonion imaginary space.
diagonals finally get identified. this seems to be abandonment of (<3)'point-of-view' for (>3)'space-of-view'.
step 5 - ??? i run out of languague-skills. is this the lost of operation properties, the running out of 4 boolean-logic values ? the sequence might be considered to end at step, 4 from this perspective. beyond would lie infinite recursivity, from another perspective.

i think i read a comment from wendy in some thread in these forums, where she says that in a certain sense, 4-space (or point?) corresponds to 2d, 5d corresponds somehow to segments and 6d is like 'physical 3-point' ? sorry, if it's not exact quote, but it clearly indicates in some experiential sense, that experiencing 4d is 3d-1d(depth i guess?), 5d: horizontal is lost (thus just 'verticality' is left), and 6d-point of view is like all the views from all the points in 'infinity' (RP3 ?) on one 3-d point ?

i made prolly perspectival grids for one view for each of these levels-of-attunement. coordinate templates, they might need some correction from someone mathematically skilled and imaginatively gifted, but i believe them to be fundamentally true... i don't have math skills, just my imagination. i believe these could really help in imagining projections of multidimensional objects as more complex and more 'mentaly-digestable' gestalts/flows. i can give sketches and references if anyone really interested.

[wendy]

There is a trick here. Normal projective rules do not apply in four dimensions, or any space other than Euclidean 3-space, because the radiant laws are different.

In four dimensions, the lines do fall off in thickness as the inverse of distance, but this makes things appear smaller as the square and cube.

In hyperbolic space (the dimension does not matter here), things fall of exponentially alarmingly fast, and things appear further away than they really are.

I put forward that the Not Knot video (of which i seen only stills), shows a poincare projection of the hyperbolic tiling {5,3,4}. People who had seen the video see the dynamics of that space, and come to associate it with the Beltrami-Klein projection.

After a good deal of discussion, we came to the conclusion that we were both right, although it took of me some convincing.

Still, the point here is that a correctly presented model of 4d or other spaces may well not "look" natural. This is because prospective is a learnt thing, and you would guage the wrong distances looking at something that is of a different set of projective rules. Still, the correct rules are better than none!.

[thigle]

here are first 4 perspectival schemes i promised, from left to right: http://tinyurl.com/9jc4u

Edit by iNVERTED: Fixed that link.

first is not a grid really. its just a representation of a bit of flat 3-space, used for most of the representations of projections of 3d or 4d or nd objects. green is for horizontals (thick is horizon itself), cyan is vertical, reds are diagonals.

second is systematized - unified view on classical flat perspective. it is representation of curvilinear pictorial space (spherical case), where circle at infinity (blue circle) is included. this perspectival margin is implicit in perspective schemes based on reduced FOV. basically, this is a 180degree FOV (fisheye) grid for frontal half-sphere of vision. FOV's radius is infinity/2.

third is the whole sphere of vision, 2pi FOV. colors are complementary for back-view half-sphere of vision, and new visual margin (=field-edge) is purple (to differ from the first (blue) infinity). this representation allows showing all views from a point in 3-space at once, it is 3-space omnidirectional-vision mode. however, it misses the viewer's locus - the point removed from projective-plane. FOV's radius is infinity/1.

fourth is a quadratic form of perspective. it is unified view of 2 whole 3-spaces, touching in left-direction_vanishing-points, nadirs, and right-direction_vanishing-points. the coloring is unfinished, as it is open to me whether they are in mirror or inverse relationship.
.
.
.

now, I was wondering if, let's say 3rd, or 4th of these would be useful for 4d learning.
for a projection from 4-space into one of its subspaces... usually, we choose one E3, and project into it. now which sub-space of hyperspace we project to is one thing, how many subspaces we project to is another. i haven't read pat's nklein manual, but i have seen snapshots from nklein, where linear sequence of projection spaces is used for simultaneous viewing of multiple slices of the same 4-object. their discreteness (apparent dis-connectedness) is annoyance to me. i believe, that a gestalt in which the projection spaces are present as a coherent whole would be much more rewarding for our curious minds.

another question is(confused one though): is the 4th coordinates-grid a view from within of an intersection of locally centered horocycle ? is hyper-klein bottle considered a horocycle?

[wendy]

A locally centred horocycle is infinitely far away, since it is the shape of the horizon. The geometry is very strange. Klein studied it as the inversion-space, eg I2, but the longsight probes out to deep infinity brought back its true nature: a horocycle is nothing more, or less, than the euclidean line.

So i guess that the euclidean coordinates would give a horocycle, since the flat euclidean plane is nothing more or less than a horosphere in euclidean space.

W

[pat]

Edit by iNVERTED: Fixed that link.

Is that link right? I'm seeing an album of 90 photos... the first four definitely don't fit the descriptions here.

[Keiji]

It was right at the time I edited it, but I can't seem to find them now either.

[thigle]

that link really doesn't work anymore. i don't have where to post images, and friend told me of that one, so i used it. i don't know what to think of it, the photos are not explicitly commented much upon in the threads i skimmed through.

if any of you know where to post images for free, please let me know and i'll put it there.

or whoever wants me to email it, please let me know via pm.

[thigle]

btw, iNVERTED, what is the meaning of the new white symbol under your name ? (i like it much more than your previous one)

and wendy, you wrote (bold added):

...locally centred horocycle...infinitely far away...is the shape of the horizon. ... studied it as the inversion-space, eg I2, but the longsight probes out to deep infinity brought back its true nature: a horocycle is ...euclidean line...
...i guess that the euclidean coordinates would give a horocycle, since the flat euclidean plane is nothing more or less than a horosphere in euclidean space.

the same again, in other format...
horocycle...
_locally centered is shape of horizon, infinitely far away
_studied by klein as I2 - inversion-space
_by longsight probes out to deep infinity(=into horocycle) seen to be euclideaan line
_given by euclidean coordinates ( as flat euclidean plane exacts horosphere in euclidean space )

now i got confused which means potentially clarified: what is the horocycle ? and what is euclidean line ? if they differ, how, if not what are their commons ?

it seems to me you say that seemingly, horocycle is somehow 'inversive' theoretically. but 'actually', through optical extension of technoempiricism, it was 'seen' to be/posses attributes of euclidean line ? did the watchers really know the pathways through the mirrorhouse ? the pluriverse is a twisted realm.


if the horizon is present here, in 'deeply infinite' abode, but at the same time it is infinitely far away, it reminds me of the following which i don't feel to grasp integrally:

en ellipse has 2 foci, definite lenght of thread from one to other, moving, generate single ellipse. now if the thread is infinitely long, the ellipse lies in infinity.
focusing by diffusion, getting infinite into picture, the distant gets close, infinite is finitized, and previous 'localness' - the 2 foci, apparently merge into 1 point: they overlap. horizon is then perceived not as ellipse but ideally as circle ?

now how is a circle to be perceived as euclidean line ? are there some other than euclidean circles ?

seen from infinitely far, the seen appears to be the seer.
seen from infinitely close, the seer appears to be the seen.
but i still don't know what is the horocycle.

[thigle]

these chapters on horocycles i find pretty instructive (i like author's modus operandi): http://www1.kcn.ne.jp/~iittoo/index.html#chapters

[wendy]

A horosphere is simply a plane of "zero curvature", that is, the result as one lets the radius go increasingly large. As long as the radius term is extremely large, the value 1/r² is effectively 0, and euclidean geometry applies.

Hyperbolic geometry has a curvature less than 0, ie 1/r² < 0, and spheric geometry has a curvature 1/r² > 0.

Regardless of the space, one can insert a subspace of curvature not less than that of all-space (eg you can draw a 1-inch circle on a 2-inch sphere, but not a 3-inch circle).

When the curvature of the subspace is the same as space, then the thing is "straight", meaning that there is only one arc of that curvature between two points.

For example, there are two ways one can draw a 1-inch arc between two points on a 2-inch sphere, but only one way [great circle] that a 2-inch circle can connect the two.

Zero-curvature then simply means that the perpendiculars to the surface converge at the horizon. For euclidean space, (which is zero-curvature), this happens when the subspace is straight or flat.

For hyperbolic space, the natural space is negative, and the zero-curvature is not flat: it is bent.

THE HORIZON

It is worth considering what the horizon is. Firstly, suppose H is a horizon point, and P a real point. With H as centre, we have a real-surface horosphere HP with a horizon-centre H. The line HP and PH are the same, and we have PH is therefore also the radius of a horosphere: that is, the shape of the horizon is a horocycle.

It isn't really one but a whole "dazzle" of them. To see what happens, imagine you have a point P, and a point H lives somewhere in a region, say 1 ft diameter. You then draw many cirles, H'P, H"P, etc. As the centre of the h circle goes further away, it is hard to separate the arcs of the circles in H'P, H"P etc.

So at the infintesimal, there are still different circles, but these are akin to adding 0 to a number: you effectively are seeing H', H", ... as one far-off point.

We now see why sometimes one can use infinity as a precise number and sometimes not. As long as you can be sure that H maps onto H' and not H", you can treat it as a fixed number. But once this certianty goes, you can't do finite differences.

Likewise, for the line HP, the absolute length does not matter, in as much as the lines H'P and H"P are seen as one, but we can distinguish between HP' and HP", because the distance P'P" is itself real (ie both ends accessable).

The horizon is essentially an angular function to real space, ie we have just direction (eg that-a-way), rather than a distance term (eg 20 miles). Since distance at vast scale is a feature of parallax, we set infinity beyond the range of parallax.

Suppose the length of an "infinite"-line is U, and the diameter of accessable space is R. As long as we can't tell the angle R/U, we can't resolve discrete points that far out. It really does not matter what R/U is absolutely equal to, it just means that it is no longer meaningful to separate it from 0.

In the Euclidean space, the radius of a sphere circumference U is ~U. That is, the real accessable space appears in the complete sphere as a point, and planes that intersect the real space pass through the centre of the sphere. This leads to the notion that the Euclidean horizon is itself the geometry S2.

In the hyperbolic space, the radius of a sphere, circumference U, is typically ln(U). This is typically of the order of R. What happens is that even small changes in the locale makes a change in the horizon: that is, the horizon changes quite considerably as one moves.

The other effect is that real lines (ie those that have an intersection with real space), make a fair intersection with the horizon. Any circle on the horizon is the intersection of a real plane and the horizon. The intersection of two isocurves is an isocurve of curvature not less than either of the intersectors, and since this is infinite, the intersection is itself a horocycle.

So, we have a isohedrix of zero curvature, in the shape of a sphere, where every circle represents a zero-curvature line (ie a horocycle). It is possible to draw an infinite number of straight lines between any two points, and also it is possible to draw a straight line between any three points.

The geometry of this surface is indeed the Inversion-geometry that Klein studied, but he did not go far enough. What happens, is that every point represents a region the size of R, and all points outside of R are on that point's horizon (remember, we have R~=ln(U)). If one inverts this sphere through the circumference of a point H on the horizon, one gets inside that point [which is really a ball the size of R], the euclidean geometry.

One then effectively has a relative metric. By supposing that H is the point of infinity, every other point maps onto by inversion, to a euclidean metric, with a defined angle and lengths. H is effectively the centre of the circle of inversion.

What Klein studied in the inversion-geometry is effectively the horizon at infinity, but he did not draw these conclusions.

Wendy