professor: all right. lecture 17 was about foldingpolygons into polyhedra, and today we will do itwith real pieces of paper. but before we getthere, i want to talk about some things we could make. in lecture, i demonstratedthis perimeter having technique where you takeany convex polygon and you pick any pointon the perimeter. then you measure out theperimeter halfway on each side.
you get the antipodal point, y. and then justglue-- locally, you glue everythingaround x and around y. so i mean, you just start. we call this zipping where youdon't glue any extra material in here, you just startzipping these guys together. i guess these circles or whenyou happen to hit this vertex, well, then you just keepgoing-- zip, zip, zip, zip, zip. so that's one thing we couldmake with convex polygons.
slightly more generalis called a pita form. this is defined in the textbook. and the idea is insteadof a convex polygon, you could take any convex body. it could have some cornersor it could be smooth. so this is whatwe call a convex, or what i'm going to calltoday a convex 2d body, meaning it can be smooth inaddition to polygonal. and then you pick anypoint on the boundary
and you measure out theperimeter halfway, x and y, and you glue inexactly the same way. now, it does not followfrom alexandrov's theorem that this will makea convex thing, but it follows from aslightly more general form of alexandrov's theorem calledalexandrov-pogorelov theorem. so pogorelov was astudent of alexandrov and he edited some thingsthat alexandrov wrote, but then also pogorelovwrote his own papers
and proved somestronger versions of alexandrov's theorem thathold for smooth bodies, not just polygonal folding. so alexandrov'stheorem, you recall, if you have a convexpolyhedral metric homeomorphic to a sphere, then it's realizedby a unique convex polyhedron. so the pogorelov's extensionis that if you have everything except the polyhedralpart-- so if you have a convex metrichomeomorphic to a sphere-- so
topologically it's a sphere,but we omit the polyhedral part. so polyhedral was that youhave finitely many vertices. in some sense, i haveinfinitely many vertices, infinitesimal curvatureat every point here. then it's realized by aunique convex 3d body. so same concept ofa convex object. well, i should say thisis really the surface of. so a convex body, i shouldprobably define that. convex body is one whereyou take any two points
and you draw astraight line and you stay within the body forthe entire straight line. so that's convex. you take any two points,like this one and this one, draw a line between them,and you stay inside. the body is the piece ofpaper we're gluing up. here we have a 3dbody, but then we just look at the surfaceof it, the boundary. and that's going to bethe surface we make.
technically, thatsurface is not convex. it's the interior that's convexbecause you can draw lines through it and you stay inside. this is exactly the same thing,but instead of polyhedron here, we have body because we didn'thave finitely many vertices. and i'm not going toprove this theorem. but existence isbasically the same proof. you just take a limit. so you can polygonalizewith lots of little edges
here, add little points,and take the limit as that approximation gets very,very close to the actual curve. at all times youhave a polyhedron, it's easy to show you willconverge to a convex body. the uniqueness is theharder part, i believe. i haven't read the proof,but it was more challenging. i think that took 20 yearsor so to settle uniqueness. alexandrov's like 1950. this theorem i think is1973 the final version.
but the version without uniquewas proved back in 1950. so that's somethingelse we can do. instead of cuttingout a polygon, we can cut out a nice smooththing or you can add some kinks and be smooth elsewhere. as long as it'sconvex, then you're guaranteed that-- we'reonly gluing together two convex points. it could be flat or itcould be strictly convex,
which means we'll have at most360 degrees of material at any point because it's alwaysdoing two at most 180's. and so we will havea convex metric, meaning always at most 360of material everywhere. so then this theorem appliesso we'll get a unique thing. another thing we couldmake is called a d-form. and i have some examplesof d-forms here. so the idea with the d-form isyou take two convex 2d bodies-- so here they have somestraight parts and some curve
parts-- of the same perimeter. i pick one point from onebody and i pick another point from the other body. i attach them there. and then i zip. so it would be zip,zip, zip, zip, zip. wherever my fingerswere at the same time, those are points thatget glued together. and this is with theconvex body you get.
again, alexandrov-pogorelovapplies because we're still only gluing two convexpoints together at any point. as long as these guyshave matching perimeter, we'll be ok. and so we will get aunique convex 3d body. in this case, we getthis fun kind of shape. so this is what isuggest you make. d-forms tend to look alittle bit cooler somehow because they have two polygons.
and the one that'sparticularly easy to build is two copies of thesame shape, but then you don't want to gluecorresponding points. like if i glued thispoint to that point, i would just get a flat,doubly covered thing and that's boring. but you just picksome other point and glue it tonon-matching points, then it acts like you havetwo very different convex
bodies which is kind of cool. i have some imagesof d-forms here. there's actually a whole bookon d-forms by john sharp. they're invented by this guytony wills who's an artist. at the top, thequote is, "there is no such thing asan ugly d-form." so we are guaranteedsuccess here. and also it says, "theysurprise, confuse, can be addictive.
they're an intellectual virus. beware." so you've been warned. this is tony wills,the inventor. now, from theartistic perspective, you don't have to startfrom convex polygons. we're going to startfrom convex bodies because they'reguaranteed to work. if you're careful to satisfyalexandrov-pogorelov,
you can have a convexmetric even though you start from nonconvex bodies,although in this case, he's actually gettingnonconvex shapes as well. more typical d-form issomething like this. this is made from two ellipsesand he builds them out of metal. this is a simulation bythis guy kenneth brakke who has this softwarecalled surface evolver. and it's usually used forcomputing minimal services,
soap bundles andthings like that. but here it's computingwhat d-forms look like, approximately. and so this is an exampleof taking two ellipses and instead of-- if youglue at matching points, they would be flat-- butthen you change the rotation. so here, they're rotated by 22.5degrees, rotated by 45 degrees, rotated by 90 degrees, soyou get this nice continuum of different things thatyou could zip together.
so i propose we make some. and so you have yourscissors and tape. if you don't have two sheets ofpaper, get some from the front. and so everyone havescissors, tape, everything? we're going to haveto share a bit. so what we want to do is maketwo identical convex shapes. so you have some here. two rectangles arevalid convex shapes. you could cut not all.
everyone should dosomething different. i'm just going to make surethey're nicely aligned. hold on tight because youdon't want them to let go, and then just cut. always turn leftwith your cutting. you can be smooth. you can have some corners,what you want to do. just always turn left. and once you have your twoshapes, you pull them apart.
make sure you do not gluecorresponding points. pick some other points andattach them together with tape. here's where it gets alittle bit challenging. with smooth curves, it's littleharder to tape them together, so you're probably going tohave to use a bunch of points of tape, so to speak,not literal points. and i just tape, tape, tape. before you get allthe way, you want to make sure you can stillpop it up and be convex.
i think we'll be able todo that by [inaudible] inside the limit. give me some scissors. all right. there's my d-form. mostly convex, kind of fun. so a funny thing you'llnotice about d-forms, if you made yourssmooth-- so if you don't have any kinks on theouter boundary-- then
the surface looks smooth,which is kind of nice. the other thing is there'sthis noticeable seam where you taped things. there, you're notsmooth, obviously. you've got a creasealong the seam. but also it looks like allthat the surface is doing is kind of takingthe convex hull, the envelope of that seam. so there's twoproperties we observe.
one is that it lookssmooth, except at the seam. and the other isthat the hull surface is the convex hullof the 3d seam. and i want to proveboth of those things. those are both true facts. but first i need todefine some things. so we're going toprove that d-forms are smooth andother good things. this is a paperthat was originally
a class project in thisclass, i think in 2007. sounds about right. submitted during class, it lookslike, or just after semester with greg price. d-form have no spurious creases. so what's a d-form? you take two, forus, it's going to be two convex shapesof equal perimeter. you glue two points, twocorresponding points p and q
together, and thenyou zip from there. we can actually talk aboutsomething even more general which we'll call a seam form. a seam form, you can havemultiple convex shapes. it's a little harderto imagine, but you can join multiplepoints together. maybe you join all threeof those corners together. as long as you have at most360 of material everywhere, it'll be fine.
maybe do some zipping. i'm not going to try to figureout exactly how this gluing pattern works, but youfind a gluing pattern where you don'tviolate anything. so it looks likeif i do this one, i better have theseguys also a little bit sharp so that they don't addup to too much angle there. so these three pointswould also join together. this would be like threepetals in like a seed pod
or something, like those leavesthat wrap around little fruits or something. i won't try to draw it but--i guess i will try to draw it. something like this. so that's a thing. this could even foldnot exactly a sphere, but something like a sphere. so seam forms, you could haveany number of convex shapes and glue them togetherhowever you want,
provided at all points yousatisfy alexandrov-pogorelov, so you only have at most360 degrees of material and you aretopologically a sphere. then you have a seam form. so most of the theorems i'mgoing to talk about apply to general seam forms, but we'reinterested in d-forms and pita forms in particular. so let me tell you sometheorems about seam forms. theorem 1 is thata seam form equals
convex hull of its seams. so the seams ingeneral are these parts where you did gluing,wherever you taped stuff. the boundaries of the convexpolygons, those are the seams. they map to some curves in 3d. if you take theconvex hull, the claim is you get the entire seam form. that's actuallypretty easy to prove. then the secondclaim is there aren't
many creases other thanthe seams of course. the seams are usuallygoing to be creased. and these creases aregoing to be line segments. and not just any linesegments, but their endpoints are pretty special. so the claim is thatthe creases have to be line segments basicallyconnecting strict vertices. what do i mean by strict vertex? something like these threepoints which come together,
provided the total angle hereis strictly less than 360. i call that a strict vertex. whereas these pointsconnecting from here to here, those don't countas strict vertices. they do have curvature, butonly infinitesimal curvature. because this is essentially180 degrees of material, just slightly lessbecause it's curved. so only where i havekinks can i potentially have strictly less than 180.
and if i join them togetherto be strictly less than 360, that's a strict vertex. so potentially, i have aseam coming from here, also at the other endpoint. so in this picture, i mighthave a crease like this, i might have acrease like this, i might have a creaselike this, potentially. but in something like a d-form,there are no strict vertices. if these are smooth-- so i'llassume here these are smooth.
we could call it a smoothd-form if you like. then there's no vertices,no strict vertices, only these sort ofbarely vertices. so there could be no creases. there's one othersituation which is creases could betangent to seams. this should seemimpossible because the way i've defined thingsit is impossible. you're tangent to a seam, thatwould be you're like this.
how am i supposed to havea crease inside that's tangent to the seam? the answer is you can't. if these regions areconvex, you can't have them. this statement is actuallyabout a more general form of seam forms where youcan have a nonconvex piece. so in general seam form, youhave a bunch of flat pieces and you somehow searchfor them together so that you satisfyalexandrov-pogorelov.
and now you couldhave a tangent. i could, for example, have acrease that emanates from there or have a crease thatemanates from there. but right now, ithas nowhere to stop. the only way for thesecreases to actually exist is if there's akink here, then this could go from there to there. so this is starting at avertex and ending tangent. you could, of course, alsostart tangent and end tangent
if there's another bend. but the claim is,all crease look like that, whichmeans if you carefully designed your thingor vaguely carefully such as a smoothd-form, then you're guaranteed there are no creaseswhich is what we observe. so this is justifying ourintuition from these examples. one other case is the pita form. so pita form doesactually make vertices.
this point x is going to bea strict vertex because it has less than 180 ofmaterial, so definitely strictly less than 360. same with y. so pita form ispotentially going to have a seam from xto y, but that's it. sorry, a crease from x to y. it has at most one crease. i think most exampleswe've made do
have such a crease from x to y. let's prove these theorems,or at least sketch the proofs. don't want to gettoo technical here. so to prove the firstpart that the seam form is convex hull of its seams, it'shelpful to have a tool here which is a theorem by minkowski. we'll call itminkowski's theorem, although he has a bunch. hermann minkowski.
it relates convexthings to convex hulls. it says, any convexbody is the convex hull of its extreme points. what are extreme points? extreme points are pointson the surface where you can touch just thatpoint with a tangent plane. so if you think ofpolyhedra, these are vertices of the polyhedra. but i want to handlethings that are smooth,
so they may might notreally have any vertices. in general, i havesome convex body. if i can draw-- imaginethis is in two dimensions-- if i can draw atangent plane that just touches at a single point,then i call that point extreme. let's look at an example here. here i believeevery seam point is going to be an extremepoint because i can put a tangent plane.
it just touches at that point. what am i distinguishing from? well, for example, this point,this surface is developable. we know it's ruled. so there's a straight line here. you can see it in theshadow, in the silhouette. if i said, is thispoint extreme? i try to put a tangentplane on, the only way to get a tangent plane isto include this entire line.
so it's impossible tojust hit this point or any point along that line. in fact, every interiorpoint, i claim, you cannot just hit that point. you've got to hitan entire line. whereas at theseam, i can do it. i can angle betweenthis and this. there's a tangentplane, because this is curving, that onlyhits at one point.
if i had a straightsegment, then the endpoints of thestraight segment of the seam would be extreme, but the pointsin the middle of the segment would not be extreme. so that's themeaning of extreme. and we're going touse this because i claim that these interiorpoints can never be extreme. it can only be the seampoints that are extreme. and therefore, we are theconvex hull of the seam points
because we are the convexhull of the extreme points. how do we argue that? so i claim an extremepoint can't be locally flat and at the same time be convex,because convex in 3d, that's what we need. if you had a point and itjust meets a tangent plane at that single point,that means locally you're kind of goingdown from that plane, if you think of thatplane as vertical.
so this is a situation. i want to prove this. to prove it, i need tointroduce another tool which is a generally goodthing for you guys to know about, sokind of an excuse to tell you about the gausssphere, which i don't think we actually cover in lectures,but might come up again. gauss sphere is a simple idea. for every tangentplane-- let's just think
of this single point. i want to look at the-- it'scalled the tangent space. look at all the tangentplanes that touch this point. because thisparticular tangent plan i drew only touches that onepoint, it has some wiggle room. i can pick any directionand just wiggle and rotate the planearound this point and it won't immediatelyhit the surface. i've got a little bit of timebefore it hits the surface.
so there's a two-dimensionalspace of tangent planes that touch just this point. because there'sat least one, i've got to have some wiggle room. so i want to draw that space. and a clean way to draw thespace is for every such plane to draw a normal vectorperpendicular to the plane, take the direction ofthat normal vector, and draw it on a sphere.
it's a unit sphere. so this is a sphere. this one looks pretty verticalso that would correspond to the north pole of the sphere. so that direction becomesa point on the sphere. think of it as this vectorfrom the center of the sphere to the north pole. but we'll justdraw it as a point. and because i've got atwo-dimensional space
of maneuverability orrotatability of this plane, i'm going to get someregion-- which maybe i should draw likethis-- of the sphere. those are all thepossible normal vectors of tangent planes at that point. so this is called the gauss mapwhere you map all these normals to the gauss sphere, which gausssphere is just a unit sphere. fun fact. the area of that thing,which is the map of all
the tangent normal directions,equals the curvature at that point. you may recall atsome point curvature is actually gaussiancurvature, so that's why gauss is allover the place here. now because i claim thisis a two-dimensional space, this thing willhave positive area. therefore, at thispoint, that vertex has positive curvatureon the surface.
and yet, it was supposedto be in the middle of one of these flat shapes. that's supposed tohave zero curvature. contradiction. or stated morepositively, if i have a point that isan extreme point, it's only touched byone of these planes, i have this flexibility. therefore, i get area.
therefore, it isnot locally flat, so it must be a seam point. it could be one like thiswhere i'm barely non-flat, but i am non-flat. there's kind ofinfinitesimal curvature here. or it could be one of thesepoints where i'm very non-flat, or i guess x and y hereare other examples of here i'm very not flat. i've only got 180 degreesof curvature roughly.
so it's got to be oneof those if you're going to be an extreme point. therefore, all the extremepoints are seam points. therefore, convex hullof the extreme points is the convex hull of the seams,and that is your seam form. so that's part one. it's pretty easy. let me tell you about part two. part two builds on part one.
it's one reasonwhy we care about. so part two is that thereare no spurious creases. unless you have strictvertices, then you could connect those up. so let's look at alocally flat crease point. actually, at this point,i should probably mention. this paper, i think of as aforerunner to the "how paper folds betweencreases" paper which is that hyperbolicparabolas don't exist.
you may recall there weretheorems about what creases look like, what ruledsurfaces look like, and so on between creases. in that setting, we've provedthings like straight creases stay straight and allthese good things. here, it's a littlebit different because we have kind oftwo levels of creases. there's the seamwhich is special. and then we're imagininghypothetical creases
between the seam. and could you havea curved crease? claim is no. claim is all the creasesin between the seams have to be straight. and in fact, theycan't look like this. they have to be atcorners somehow. so that's what we'regoing to prove. but a lot of thesame techniques,
most of the same definitions. this paper waskind of a warm up, i feel like, forthe nonexistence of hyperbolic parabolas. although they use alot of the same tools. they have two ofthe same authors, but there's notheorem that really is shared between the two. they're not identical.
so we're lookinginside a flat region. we're imagining, let'slook at a crease point. it lies on some crease locally. and it's a fold crease. so i'm going to wavemy hands a little bit. but imagine a folded crease. maybe it's a curve. we don't know. it's folded by someangle, some nonzero angle.
otherwise, it's not a crease. so locally, it looks liketwo kind of planes, at least to the first order they'regoing to be planes. maybe something like this. i'm going to drawthe creases straight because to the firstorder, it is straight. but it might be slightly curved. so we're lookingat a point here. and my point is, it'sbent by some angle.
so i'm going to drawthe two tangent planes. there's a tangentplane on the right. whatever the surfaceis doing on the right, there's a tangentplane over there. whatever the surface is doingon the left of the crease, there's a tangent plane there. they're not the same planebecause we are creased. whatever the creaseangle is, that's the angle betweenthese two planes.
i want to look at this tangentspace again a little bit. you can think of it,the tangent space at the least it hasthis tangent plane. for this point, you haveat least this tangent plane and at least thisother tangent plane. and so in particular, you haveto have the sweep between them. so you get at least aone-dimensional set there. now, this can happen. so we're not going toget a contradiction.
we're not going tosuddenly discover there's a two-dimensional areaand get that it wasn't flat. but at least we've gota one-dimensional arc. on the gaussian sphere here,we've got something like this so far. this is one extremetangent plane on the left. this is extreme tangentplane on the right. these two pointscorrespond to those two. and then we've gotthe arc of-- you
can sweep the tangentplane around and still stay outside the surface, well, that's interesting. it's interesting becauseall of those tangent planes share a line, which is going tobe this line that i drew here. i mean, you look atthese two planes, there's a line that they share. and if you rotate theplane through that line, you'll still passthrough this point
and you won't hit the surface. so in fact, all those tangentplanes share this line. i think we actuallyjust need these two. so if you don't seethat, don't worry. there are these two planes. there's a line there. they're tangent,which means that line is on or outside the surfacebecause tangent planes don't hit the surface.
i mean, they touchthe surface barely, but they don't go interior. well, i claim that in fact thesurface must touch this line. i claim the surface,at least locally, has got to followalong that intersection line of the left andright tangent planes. why? it comes down to thisgaussian sphere thing. we know that the surfacelies inside these two planes.
there's this wedge here. the surface must be below. we also know the surface comesand touches it at this point. could it, from here, kind ofdip down away from this line? i claim no. if it dipped down, then there'dbe a third tangent plane. the tangent plane couldfollow and dip down as well. and then we've got notjust a one-dimensional arc, but we're actually going to geta whole triangle in the gauss
sphere. once you have a whole triangle,that means you weren't flat. you have positive curvature. so we're looking at alocally flat crease point somewhere in the middleof one of these polygons. so in fact, you can't affordto dip down because then you'd have at least a littlebit of curvature there. because you have zero curvature,you've got to stay straight. so that means thatthe surface locally
has to exist along thesegment in both directions until something happens. now, we assumedthat our point was locally flat and a crease point. so the surface mustcontinue in this direction until we reach a point thatis either not locally flat or not a crease point. i claim it's got toremain a crease point. because look, you're followingalong this intersection
line of these twotangent planes. these remain tangent planes. these are not only tangentplanes at this point. they're also tangentplanes at this point and potentially anypoint along this segment. as long as the surface ishere, they are tangent planes. therefore, you are creased here. i mean, you can't besmooth if you're butting up against these twotangent planes.
so crease must be preserved, sothe only issue is locally flat. it could be at some pointyou become not locally flat. and that indeed happens. that's when you hit the seam. when you hit the seam, thenyou're no longer locally flat. so this proves thecreases must be line segments betweentwo points on the seam. there's a little bit more toprove which is it either hits a strict vertex orit's tangent to seams.
and this basicallyfollows the same argument that if you hit something otherthan a strict vertex or a seam, you would have positivecurvature where you shouldn't, basically. so i'll just leave it at that. i think that'senough for the proof. you get the idea whycreases can't exist. any questions about that? i have few more questions.
one question is, can we seemore examples of rolling belts? and in some sense,this example is an example of a rolling belt. so d-forms, you taketwo, say, ellipses. and the seam that connectsthem is a rolling belt. and this is an illustrationof the rolling action. as you change, which pointglues to which point? let's say i fix this point. i can glue to this one orto this one or to this one
or to this one. and then i zip around. that's the rolling of the belt. more generally,i'll draw a picture. more generally, a rollingbelt from the viewpoint of a gluing tree is thatyou have basically it's like a conveyor beltlike at the airport or on a tank or something. you've got treads hereand it's going to roll.
so if i mark a point, let'ssay i mark this point. and i roll it aroundthis way a little bit, this point maybestays stationary and i end up withsomething like this. and what that means is whateverthis guy glued to over here is now over here. so we're changing the gluing. this guy's now getting gluedto some point like this because it kind of rolls around.
so in general,you're rolling this, but then you're always justgluing across this path. for this to work, every angleout here must be less than or equal to 180,like the material that's out on the outside. remember, the gluingtree has the outside of the polygon in here. the inside ofpolygon's out here. so all the material'son the outside here.
all of these things must beless than or equal to 180. if you can find a pathin your gluing tree where you've got less than180 material all around you, then you can do this rolling. and no matter how you roll,and then just glue across, you will still have a validgluing, a valid alexandrov gluing. so that's what rollingbelts look like in general. we see them all over theplace with these d-forms.
the simple examplei showed in class was when you take a rectangle,you glue into a cylinder. and now up here, i'vegot a rolling belt. because if you lookat the sides here, i've got actually exactly180 degrees of material all around the belt. and so i couldcollapse it like this or i could collapseit-- let me tape it. a little hard tohold everything.
if you look at this belt, icould collapse it like this. or i could roll it a littlebit and classify like that. or i could roll it a littlebit, collapse it like that. roll it a little more. collapse it like that. all of these are validgluing because they're always gluing 180 to 180. and that's whatrolling belts are. does the broken applet work now?
i don't know whyit wasn't working. probably some conflictwith suspension or just the javainstallation was bad. but here it is. it's freely available online. you draw your graph. it's a little hard to just use. it's hard to drawan exciting example. here i'll a draw a tetrahedronbecause that's simple.
in this case, it's abstract,so all of these edges are unit lengths. it's kind of like treemaker. you separatelyspecify the lengths. then you say compute. and it will find, in thiscase, it's a tetrahedron, an equilateral. a regular tetrahedronis a little hard to see because we've goteither a weird field of view
or a lot of perspective here. but this is aregular tetrahedron. you can verify that by saymaking some of these lengths longer. i'll make three of them five. so then this is like a pyramid. of course, we're justessentially applying cauchy's rigidity theorem hereand say, oh, this uniquely assembles into thisspiky tetrahedron.
you could, in principle,drawn an across here with the appropriate gluing. it's a little trickyto draw in because you have to also triangulate. it'd be hard to do fromscratch to compute gluings. but in theory, especiallyif you use software to compute the gluingsand compute shortest path in a gluedpolygon-- which we'll be getting more to next class.
next lecture is aboutalgorithms for finding gluings. then computing shortest paths. then you could use that asinput to this software which will then compute whatthe 3d polyhedron is. so that's the alexandrovimplementation. this is by, i think, a studentof bobenko and izmestiev. one quick questionabout-- running out of time-- aboutthe nonconvex case. so we're doing all this workabout making convex surfaces.
i said nonconvexcase is trivial. in a sense, the decisionproblem is trivial. it's not an easytheorem to prove. and aha, taped it up. it follows from what's calledthe burago-zalgaller theorem. and it says that if you tapany polyhedral metric-- so that means you take a polygon,you glue it however you want. you could even add boundary. pick any polyhedron metrics, soyou take any polygon and glue
it however you want. it could have handles. it could be a doughnut. it could be whatever. then it has an isometricpolyhedral realization in 3d. so there is a way-- sowe're doing it abstractly. we're gluing stuff together. it's kind of topologically. we don't know what we're doing.
but then there is a wayto actually embed it as a polyhedral surfacein 3d without stretching any of the lengths. so it's a folding ofthe piece of paper. it could have creases init, will have actually tons of creases in it. and it has exactly theright-- it connects all the things you wantit to connect together. it's kind of crazy.
furthermore, it couldbe self-crossing. but if your surfaceis either orientable-- so it doesn't have any crosscaps or mobius strips in it-- or it has boundary,then there's even a way to do without crossings. so that's basically allthe cases we care about. you could glue this togetherinto a sphere or into a disk or into a torus ortwo-handled torus or whatever. no matter what youdo, there is a way
to embed it in 3d as apresumably nonconvex very creased surface. this is a hard to prove. if you're familiarwith nash's embedding theorem about embeddingriemannian surfaces, it uses a bunch of techniquesfrom there called spiraling perturbations,which i don't know. and one description i readcalls the resulting surfaces "strongly corrugated," whichi think means tons of creases.
it is finitely many creases. that's the theorem. there's no bound on them. as far as i know,there's no algorithm to compute this thing. i've never seen a pictureof them, otherwise, i'd show it to you. so lots of open problemsin making this real. but at least the decisionproblem is kind of boring.
you do any gluing. it will make a nonconvex thing. to find that thing isstill pretty interesting question though. that's it.