Wonderful, Tayfun, and I know you are busy at the moment, but would you have time to add an extra short rib (parallel to the "main" short rib) at the quarterpoint to the left (as we look at your image) of the main short rib? Reference lines for this new short rib on the two long ribs will be helpful too.
Hope that's not too much to ask. Having assembled several of the maps, I think this extra rib will be a great help in getting the Shoemaker region to hold its shape.
Thanks

Well, assuming you mean the ability to look at problems from outside the usual box, I plead guilty, and cravenly point the blame at having when back in grammar school specialized in play.

Or perhaps you mean the lucky magic that constant-scale natural boundary maps fold to replicate their originating object. I trace that idea to Albrecht Durer c1519, although that's a coincidental trace I owe entirely to the insights of the art historian Erwin Panofsky and his pal Marston Morse. I didn't uncover that connection to Durer's "prototopological" mappings until well into this project. Tell the truth, I had the good fortune as a first-grader to be challenged to "fix the problem with all these world maps"; problem being, "they're all stretchy at the edges."

And as for genius, I think it goes in greater measure to the slow-witted, like me. It took a good nine years of making these maps before it dawned on me that their essential characteristic is to rigorously put "constant-scale" (the equator on a Mercator or simple cylindrical map) at the edge -- edge no more stretchy, just what my challenger asked for! The "natural boundary" requirement is merely gravy, ensuring that the map's various and sundry lobes are composed of districts that make sense -- in this case valleys (aka basins or dales) -- rather than and abstract (pragmatic . . . objective) triangles or gores.

The general trick, seems to me, is to be able without prejudice to name something new. (For more info, see M. Faraday's confessions on his discovery of electromotive force.)

Here is the foldable Ida with the Tayfun-corrected ribs:
I haven't yet had time to try putting it together, so the caveat still holds about the ribs need shaving where the folded object differs from Ida's true shape. Also the caveat that I may have failed to maintain constant-scale at map edge, probably near point "n."

Okay, so borrowing a little from both systems, here is the csnb map of Ida along with a set of three (only three!) interlocking ribs, one rib for each major axis. The ribs' white lines indicate where slots are to be cut, black lines to remain as assembling guides. (Note that the short rib will go in as two separate pieces.)
DON'T, anyone, at the risk of extreme frustration!, take this plate for anything but a suggestion. I made the rib shapes from Tayfun's ORTHO VIEWS plate, and Ida is so extremely concave that the "short vertical rib" is only a guess for accurate, because its perfect shape is hidden from both directions. It needs some uncertain amount of trimming near point z. Plus, I haven't tried assembling this yet to see if in fact I've got the ribs and the map at the same scale.
BUT, with this as a guide, Tayfun will be able to generate these three ribs easily and accurately; and thickness of slice won't be an issue.

Just to lessen confusion, the ribs shown in Ortho Views are flipped from what is shown in the map plate.

Tayfun, if you'll make them for our asteroid set so far (Eros Ida, Phobos, Deimos), I'll set each in scale with the csnb maps, and trim them in where the folded csnb-form deviates from true shape. (Eros in particular is the troublesome one to hold up shape in hollow folded paper, and probably will need another "short" rib, parallel to the main short rib and centered on shoemaker.)

The assembled result will still be a bit susceptible to dents, but the overall skills required won't be at the level of a professional model builder to get it done.

Well, here's what I think: cutting all those slices out of something looks like a real pain in the old asteroid, so to speak; and even if I had a laser cutter and matched sheet thickness to the appropriate scale, I'm sure I have the technical wherewithal to glue them all together evenly.
Surely that's not how you made (I'll stick with Ida as topic, but same applies to Phobos) this model, right? However this got made is the way to go, for good solid 3D models.
Even the original Eros models (the NEAR team had) used the assembled-slice method only to make a master object, then it was in turn used to make a mold, from which the production run was cast.

And yet all these methods leave us wanting the photomosaic imagery in real space, not just in the computer where we can't get our hands on it. Whence the advantage of foldable constant-scale natural boundary maps, with the disadvantage they're so easy to dent, if not crush.
So see next post for the rest of what I've come up with.

Well, so far so good. I downloaded and unzipped the file, then imported it into VectorWorks, where it appears to be all there.
I can probably figure out how to core an alignment hole or two through them, but I'm not sure I have the strength to print out each layer (must be thirty or so, right) on stock (how thick)?
Maybe someone else has a laser cutter?
Let me sleep on this. Maybe I'll think of something . . .

Great shot, by the way, Tayfun of the Northerly view of Ida. I'll work up a plate on that while I ponder this slices business.

this may surprise you but I don't have much familiarity with those programs. DXF is the only acronym I recognize, and VectorWorks is the cad system i use for architectural drawing.

Thanks, Tayfun. Like I say, the view from the other direction is not so perfect -- I have an Adkins digital pic of that side, can you send me a digital image of the shape model from the same view? I'll make another comparison plate. Here, I'll post what I have . . .

Due to Ida's irregular shape slices will not be contiguous and I suggest first we try it with Phobos. I will post the slices in DXF and CDR formats in a few days.
[/quote] I agree. Especially as Ida is the most complex shape, best to work up to it.
Have you tried to assemble the Phobos map-model yet? Phobos model hold their shape fairly well without an infill, but they'll be much stronger with it.
Note that the slices will have to be truncated, primarily at the "blunt" end, where the map precis (the choice of boundary tree) leaves the largest lobe, or flattened facet. We'll probably have to work this out ad hoc, by trial and error. At the DPS meeting, some education experts were excited about the possibilities -- tactile, hands-on activity, etc. -- but immediately asked about some kind of solid fill. So I think it's worth the effort. We have stuff here called foam-core board; I'll probably use that to cut the slices. It comes in 1/8 inch thickness increments.

Well, given the talk about South Pole Aitkin Basin on the Moon, here is a constant-scale natural boundary map of Moon the uses as its edge a network of fracture lines radiating outward from SPA Basin. (P.E. Clark picked these lines as the map precis.) The dark purple line is the basin edge. Folds up to a pretty good globe, makes a pretty seasonal ornament and, if you need it, you can pretend you hung the moon.
Kidding aside, you can see the SPA Basin-size directly, no distortion (with interruption in map-mode).
Handy cylindrical format shows same data and helps keep you oriented.
Yellow line is something (Don Wilhelm?) called Gargantua Crater.
Not to start any ruckus, but it's a lot bigger than SPA. Wonder how it compares in size to the Northern plains of Mars.

OK, I corralled a photographer, (S. Adkins) to take a pic of the folded map.

Here (on the left, courtesy of T. Oner)) is a digital image of Ida; and (on the right) the Adkins photo, taken from approximately the same point of view as the digital image.

On the other hand, I think jumpjack is best to wait and hope that Tayfun will make a slices-infill. Two reasons:

1) IN GENERAL: These are experimental maps; folding up to make a convenient, holdable model is a nice property, but the maps are far from perfect (the technology at the moment uses -- to make outlines and graticles -- the very latest of Renaissance graphical techniques, and Photoshop to add the photomosaics).

2) IN SPECIFICS: With the exception of a few crania I mapped a while back, Ida is the most complicated form yet attempted in constant-scale natural boundary mapping, certainly the most extravagantly shaped asteroid. Something (on the back side of this view!) is a little amiss -- to kinked or cramped -- around point "n.". It's easy for the geometry to slip a bit, and the edge get a little out of alignment or true constant-scale; this is probably the problem. Printers can also be a bit out of relative proportional accuracy.

So best to take this all with a grain of salt.

jumpjack:
Here is a link Phobos Arts and Crafts that gives tips and pics on assembling these paper models -- Note the link deals with Phobos not Ida, but the assembly tips ought to be transferable. Emily's article also includes (I'll have to confirm this) another link to a post I made here at USF with further tips and alternate method (clear tape) of holding it together -- Emily prefers glue.

Oh, Tayfun, now I see your post about the internal slice-packing -- What a superb idea! If you check out the latest posts (and maps) at the "Map of Eros" thread, you'll see my frustration at getting the Eros folded-up map to hold its shape in the Shoemaker region; an internal mass , even if was a fairly loose fit, would, I think, do the trick.

The problem will be to coordinate the size properly, so that the photomosaic paper will wrap the insides neatly. The "Tayfun slices" (aim for about 3/8 inch thick; keep it extremely coarse -- not many slices will be needed; as well as a bit undersized) will make on object of a certain scale, and if that matches the scale bar included on the csnb (constant-scale natural boundary) map, things ought to work. Or be close enough that a little fiddling will be all that's needed.

When I print the Ida and Eros maps on ledger paper, the assembled (folded-up) objects are about 5.5 inches in longest dimension. They aren't in front of me at the moment, but when I get back to my asteroid lab I'll put the micrometers on them and report.

I've put it together twice -- I'll post a digital pic as soon as I can find a pal with a digital camera -- and the first one went together easier than the other. So it seems that where you start, and which sides you leave the tabs on, makes a difference in the result.
The bigger size definitely helps. I printed it on 11 x 17 (ledger-size) paper.

Here is a constant-scale natural boundary map of Ida that folds to a decent replica of the asteroid. Map edges are Ida's major ridges.[attachment=16299:Ida_MaxRidge_post.jpg]
[EDIT] Here it is again with the photomosaic properly credited; apologies to M. Nyrtsov for my oversight.

I tried in Google Earth, setting sun shadows very low, but no pattern emerged, probably because contour height is not exaggerated as it is in the stretched-plastic quad sheets.

Maybe, but I am not sure. I know there is one not too far from here (Middlesboro, KY is in it).
Ted, this one, if it is one, is NOT the one in Kentucky. This one is roughly centered where NC, TN and GA come together.

I took those stressed-plastic (topography) quad sheets for the southern U.S. and shined a sharp, oblique spotlight.
One fellow told me to make an expedition to the (apparent, presumed) rebound cone and look for shattercones, but that's a bit much for an old guy like me.
RATS! My image is 1.2 megs, I'll add the post in the AM. A link is here
EDIT OK here is the image. The rebound cone is called Paynes Mountain, and Pack Mountain, a little east of Copperhill TN.

[quote name='lyford' date='Sep 10 2008, 08:17 PM' post='125720']
I hope you didn't take my comment as meaning to impugn your hard work, Chuck, which looks great - it was directed at the face on Mars crowd, if you know what I mean.

Oh absolutely. We're on the same page. The only part that curdles my milk is, apparently, what bugs you, too -- when childlike wonder and innocent imagination are co-opted by folks who need to make a movie. Besides, maybe a Saturday morning cartoon show with cutely animated asteroids saving the solar system wouldn't be such a bad idea -- but I think I know what you mean.

Tee hee! Yes, above all else, the asteroid belt -- like the rest of the universe and the figments of our imagination -- is there to amuse us.

Maybe the map of Ida will turn out like Wile E Coyote; or maybe Gus, that little mouse in Cinderella that Robert Frost was so fond of.

Here is the new constant-scale natural boundary map of Eros, showing Phil's new photomosaic.
Edges are primary ridges, the lobes come together at Eros's blunt end, per P.E. Clark's preferred precis.
We'll be presenting a poster of this and other CSNB maps at the Division of Planetary Sciences of American Astronomical Society in Ithaca this October.

EDIT: Oops, my mistake. I see that DPS/AAS results are embargoed until meeting time. What I get for being a non-member second author and this all is not in my usual field of endeavor, architecture. Sorry. I'll post this again in mid-October. MAKE THAT LATE OCTOBER!
In the meantime, if anyone wants to give a try to working out a good arrangement of tap locations (for map folding up--see discussion below; some fiddling is needed to perfect the tabs), send me an email and I'll forward a beta-image.

It cut and folds to a pretty good model, I'm claiming, though I haven't folded this up since I put the photomosaic onto it (I did a small version to check graticle matchlines), but I'll do that now -- I'm going to try it on 11x17 paper. EDIT: I tried it on paper; not so good still on holding shape in Shoemaker; looks like next I'll try 11x17 CARDSTOCK. And, perhaps, an inner cross-sectional rib might be needed.
For those who want to give it a try I recommend hunting around here (under New Phobos Names) for assembly tips; this one is a touch trickier to put together than the Phobos and Deimos maps were, but at least you'll get to hold the object in your hand. I'm also toying with which sides to put the tabs on; my test assembly didn't hold its shape in Shoemaker to well.


I've finally got a blog up, where I'll post other materials and images concerning this map; other maps are here

Yeah, that's the amazing thing to me: that an actual object in hand is a very different sensation than seeing the object on a computer screen. It's why I'm looking forward to the photomosaic folded map-model.
Too bad we don't have a kinesthesia (sense of touch) expert to weigh in with authority on this phenomenon.

Here's the current state of the last section of the Eros mosaic. Roughly 60N to 60S, 0 longitude (right edge) to 120 longitude (left edge) - but with a bit of overlap outside those boundaries so everything fits perfectly when assembled into a global map later.

Phil

Looks good, Phil; keep me posted on when the final is wrapped up. I've gotten one new constant-scale natural boundary map of Eros (outline only)of the drawing board; the "peeled banana" you suggested, but test folding it shows it, the outline, needs work. Normally I'd overlook these relatively small discrepancies in the folded model, but the whole point of this new Eros CSNB map is to make a very accurate model with photomosaic, I'm gonna try to iron the bugs before I go and spent the time loading in the mosaic. Rather only do that once. Might have something by the end of next week.

Deimos bound by ridge:

Deimos, bound by valley:

and here's Phobos bound by ridge:

Here in a series of email are the other constant-scale natural boundary maps of Phobos and Deimos. I'll describe the particulars of map boundary with each post, but all will be more conventionally map-like than the above CSNB maps, that is, these will be compact maps (without extensive lobes). Thus, unlike the above highly interrupted maps, these are more usable as maps.

The organizing principle of this series was the proposition of P. E. Clark, who seeks a methodical approach to asteroid categorization and wonders if a methodical approach to making CSNB maps will lead to same. In each case the central feature, the mid-region of each map, is the object's blunt end (the 180 degree meridian). The boundary of each map is variously a ridge or a valley; specifically the highest ridge or lowest valley, each system sharing a common end point, a saddle, specifically the object's two primary saddles.

Phobos and Deimos are captured asteroids, thus like our moon they have a near side and a far side. This has a great impact (sorry) on morphology. Sorry I
didn't realize this soon enough to mark out on these maps which side is which. I'll update the maps when I get time, but -- more fun, and maybe more useful -- I ought to redesign these maps with that characteristic in mind.

Note that CSNB mapping gives me the freedom to design and redesign the map based on physics and observable characteristics, without bothering with the traditional and usually not vary satisfactory rigmarole of abstract mathematical projection formula. Handy feature, no?

Asteroids, by the way, in their topography are particularly good examples, admirably simple examples, of minimal systems of critical points, the peaks pits and saddles (and the numerical relationship between them) worked out by J Clerk Maxwell in his classic paper of 1875, Hills and Dales. Nice, how these asteroids have two peaks (the extremes of equatorial rotation), two pits (the poles), and two saddles (where ridges and valleys intersect).


Having said that, let me digress to point out that these compact CSNB maps, unlike conventional compact maps, still have the "I'm foldable" property. They just don't fold to anything that looks like Phobos or Deimos. I've now made enough of these CSNB maps, and the consequent folded forms, to begin to see trends:
maps made from a trees with only two ends (the line of interruption is a single line; no branches) fold to apple turnoverish objects; maps made from trees with more than two ends (the line of interruption is a branching tree) -- when the number of ends is small, i.e., three or four or five -- fold to objects which resemble irregular platonic solids with the corresponding number of points.

When the tree has many branches and the branch-system does NOT spread extensively around the object, the folded form is, to crib a term from Isaac Azimov, an "essentially globular object" with fingers. But when the tree has many branches AND the branch-system spreads extensively around the object the folded form resembles the object. The more branches and the more extensive the system, the better the folded form resembles the object. Thus the maps above are good (but not perfect; more about this in a minute) representations of the respective moons.

Nothing too surprising about that last example. The metric parameter of constant-scale at the edge, and the physical parameter of the edge being a natural boundary insures the result. But to me these folded forms made from CSNB maps made from trees with low numbers of branch ends are a delightful puzzle. What's going on? Anyone who has folded up these maps surely noted, when beginning folding a branch-end, the similarity to folding a cone; then is each map an ensemble of cones? Or perhaps it's a just clever derivative of Albrecht Durer's unfolded platonic solids? Or maybe it's a profound extension of Albrecht Durer's unfolded platonic solids, to use the art historian Erwin Panofsky's term for it, "prototopological" cartography? Maybe it's something new in mathematics to think about, or maybe I'm just fascinated that I can develop forms without using any customary orthography or perspective (ray trace) point-plotting.

Think about this:
In the case of the two-end tree, the one that generates maps that fold to apple turnovers (the valley-bound maps):
If the edge were to lose its azimuth preservation, its wigglyness, the map would be a circle with the same perimeter as the wiggly map (remember the edge is constant-scale, so it can't stretch), the folding points would be diametrically opposite and the folded form would be a flat semicircle (zero volume). We all know that given a constant length, the circular shape encloses the largest area. So the wiggly map fits inside the circular map; the two dimensional area of the intersection of the two maps is (somehow!) an indicator of the apple turnover's volume.

I've no insights yet into the higher order forms.
Any of this of interest to anybody?


Well, that's the digression. Here's Phobos, bound by valley (Clark's choice was that portion of valley to the south of the primary saddles):

Yes, Emily is not only a good writer -- she taught me the term "seam slice"-- she is also adept at handcraft; I've yet to try assembling one with glue, I'll bet it improves the appearance. Emily's suggestions for which side to leave the tab are also spot on.
I heard she was here in Houston at the start of the week, but I didn't get a chance to thank her for the nice article.

Here is the companion cut'n'fold map of Deimos.

My sister has me put a little bent-wire hanger into a small hole poked into the folded form's north pole and she uses them each december to decorate the christmas tree. A little more work and we'll have a complete solar system.
Phil Stooke says I'm to do Ida next.

Asfor the other, more compactly arranged maps of Phobos and Deimos (only one is posted here so far (above, also of Phobos), but there is also a valley-edged map (the thematic inverse of the Ridge-edged map), and similar for Deimos.

The interesting thing, to me at least, is that these compact maps also fold up; not to good representations of the originating objects but rather to quite curious forms not unlike an apple turnover (the valley edged maps) and a conic-ish tetrahedroid. I don't know what to make of these curious forms, but when was the last time something novel arose in mathematics -- this I claim for world maps with constant-scale natural boundaries -- and it failed to be meaningful?

All in all, though, it's quite encouraging to see the interest in these foldable asteroid maps.
Thank you all.
Chuck