IC16:3D-Xplormath: More Than 300 Mathematical Objects, All With Animations And Explanations.

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IC16:3D-Xplormath: More Than 300 Mathematical Objects, All With Animations And Explanations.
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The Java version of the free visualization program 3D-XplorMath has long been part of IMAGINARY. I will demo the Pascal version (unfortunately still MacIntosh only) which now has explanations for all its objects. It is also richer in animations which deform the objects or add visualizations of the object‘s construction. All 3-dimensional objects can be shown in anaglyph stereo, red-green glasses will be provided. The program is intended for individual experimentation. Its development has mainly benefitted from demo-lectures to audiences ranging from school kids (13 years and up), students of my courses and colleagues at conference evenings.
Area Mathematics Mereology Imaginary number Local Group
Logical constant Surface Link (knot theory) Pascal's triangle Multiplication sign Weight Surface Counting Power (physics) Mathematics Category of being Function (mathematics) Lipschitz-Stetigkeit Electric current
Category of being Surface
Point (geometry) Surface Octahedron Kepler conjecture Conformal map 1 (number) Curve Ordinary differential equation Icosahedron Wave Cube Object (grammar) Spacetime Dodekaeder Local ring Category of being Axiom of choice Link (knot theory) Fraktalgeometrie Chaos (cosmogony) Group action Tetraeder Mathematics Plane (geometry) Object (grammar) Abelian category
Existential quantification Length
Point (geometry) Dot product Plane (geometry) Length Queue (abstract data type)
Plane (geometry) Cube Queue (abstract data type) Position operator
Coefficient of determination Greatest element Cube Boundary value problem Position operator
Standard deviation Green's function Numerical analysis
Position operator
Voting Multiplication sign Projective plane Graph coloring
Point (geometry) Rotation Dot product Dependent and independent variables Mathematics Velocity Direction (geometry) Curve Circle Planar graph Resultant Tangent
Rotation Point (geometry) Dot product Randomization Plane (geometry) Velocity Curve Circle Parameter (computer programming) Cartesian coordinate system Sphere Tangent
Point (geometry) Radius Symmetry (physics) Triangle Circle Summierbarkeit Line (geometry) Distance Ellipse Maß <Mathematik>
Point (geometry) Dot product Randomization Great circle Multiplication sign Curve Line (geometry) Distance Sphere Planar graph Ellipse Wave packet Category of being Mathematics Radius Plane (geometry) Symmetry (physics) Triangle Summierbarkeit Circle Spacetime
Torus Point (geometry) Dot product Coefficient of determination Moment (mathematics) Green's function Knot Physical system
Different (Kate Ryan album) Phase transition Moment (mathematics)
Statistical hypothesis testing Point (geometry) Logical constant Functional (mathematics) Graph (mathematics) Multiplication sign Surface Direction (geometry) Curve Maxima and minima Grothendieck topology Curvature Radius Cylinder (geometry) Helix Geschlossene Mannigfaltigkeit Circle Space group Nichtlineares Gleichungssystem Resultant Mathematical optimization Position operator Condition number
Point (geometry) Family of curves Group action Scaling (geometry) Divisor State of matter Curve Parameter (computer programming) Limit (category theory) Approximation Tessellation Equilateral triangle Space-filling curve Mathematics Many-sorted logic Cube Boundary value problem Iteration Spacetime
Mathematics Theory of relativity Surface Curve Gastropod shell Circle Parameter (computer programming) Line (geometry) Sequence Family Event horizon
Minimal surface
Group action Surface Multiplication sign Parameter (computer programming) Mortality rate Rectangle Graph coloring Mathematics Minimal surface Angle Well-formed formula Green's function Descriptive statistics
Functional (mathematics) Group action Dot product Many-sorted logic Mass flow rate Surface Combinatory logic Curve Maxima and minima Mortality rate Distance
Rule of inference
Trail Distribution (mathematics) Surface Multiplication sign Numerical analysis
Coefficient of determination Many-sorted logic Different (Kate Ryan album) Multiplication sign
Geometry Point (geometry) Dot product Many-sorted logic Surface Multiplication sign Image resolution Mathematical analysis Object (grammar) Spacetime Power (physics)
Point (geometry) Imaginary number
Point (geometry) Functional (mathematics) Randomization Numerical digit Distribution (mathematics) Curvature Quantum state Gradient Moment (mathematics)
Point (geometry) Partial differential equation Dot product Distribution (mathematics) Projective plane Student's t-test Connected space
Point (geometry) Mathematics Minimal surface Extension (kinesiology) Regular graph Imaginary number
who hold the area of the world and the 1st so again good morning and and for each all the all of you in particular since I learned yesterday that the the yellow version which is really part of the imaginary runs no longer and this is the old page of all programs
and unfortunately it only runs on Macintosh so far we had 1 desk already and computer was from Holland saved us when they're Apple terminated was that time and we are hoping that by the end of this year maybe he will have our program running and the next the so we'll see so this is the home page and the download the other and I think that's
easy enough you don't have to install anything the program comes in the full and you simply open so I believe the indentation is connected with the Corporation and with young count on surfaces of constant weights and fortunately surfaces of constant with it how the simplest surface property which I know all of this can be explained people and therefore we only have this 1 Matt lips and since some of you said yesterday that they could see cross-eyed but what I would like you to try this because if you can see it cross-eyed is really great
so the last thing I do not think this thing about the book and that and that this is not a lot to be by the the on the the what it was the
worse the trouble of course because I know of no other such simple surface properties or simplest examples of what he drowned in 1st of back this and the programmers have some partitioned into the use of
objects and the simplest ones of around and 1 point I have
entertained 7th graders 90 of them for 3 hours showing them how these point he drowned made for example in this way the no you have
to fix the length of this edge which is growing on the top side and something's not seen better and wider frame and also now you can use your the the and when and
this this fire or is it that way it was found that the head and the head of the
map room upon his heads of head and the edge length of
the queue of the edge length of the queue is the vertical diagonal in off the Pentagon or
came out now my the because there are these youngsters we have more simple things like the intersections of the plane and I guess I need think of points so you can you see the dots you should see a cue
sticking through a plane status of
OK now 1st of all we can rotate this and you see these
intersections well the and we
can change the position of the queue relative to the plane and see the different
intersections and then of course these youngsters cannibalism for 3 hours so in the documentation we have
prepared instructions to do on leaves by hand new so 1st of all to see the relative position of these dogs a relative
to those my it's much easier if there's a black boundary around them so the bottom the more is better than the top 1 and then now you have to start to do this really by hand and be surprised how well it
works and then finally so
that they could get themselves to work their hands which worked with these gasses the red is very easy but for the green you have to search and there there I know the 5 standard numbers off green pain which will work with these gasses in OK and how to use them for example the tetra hidden is in this way now they are a little bit too far apart for the glasses but if you use the grasses and then do this cross-eyed trick they would jump to something way ahead of the
screen OK and then have I was
really fascinated when it's old pictures of some of these snaps 20 he dropped and at 1st I couldn't really understand what they look like there are how you would make them but if you see a deformation running then it's very easy the Pentagon gets a little bit smaller and a little bit rotated and then when you make it small enough and rotated in the right position then you end up with uh the snap put he dropped and I find it quite surprising uh how she knew this invented all these things OK these are from the you this more than for example the mole and on how an eye piece of work In
votes and they have for some reading on the user but we don't know where that works anyway the In this way and we tried to explain you have 1 projection used to projection to change the colors of the 2 projections and the thing thing jumps into the screen when really need some time to look at that but I don't want to spend this
now OK after the 20 he drowned the
next simple thing about the planar curves there will be a little bit short on those highlighted like in the ABC quotes particularly because they go back to what in fact most of the curves which have names go back to the Greeks so that's not special to these and what you see is a mechanical construction which draws the curve and I believe you can see the little blue tangent and how it's made and I also want to explain how this comes about by the blue dots which are rotating big enough for should make them bigger have no response OK let's make them bigger in yeah the OK what you should see is that the rotation speed of these dots becomes larger as you go away from the point of contact and actually actually these velocity vectors they fit together with circles and act as a true result every momentarily rotation about the point where you have a velocity is 0 every velocity vector looks exactly the same as if you had a precise rotation around that point and therefore the tension construction which works for the circle the you connect the point at the center so the point we don't have a rotation to the point on the circle and take the orthogonal direction and that's the change and that works for all these constructions now in some cases it's more difficult to find the point the center point for these local stations but what
I like is that the same thing can be done on this year so this is
the spherical circle at and it is made exactly the same way and every argument which you just said can be repeated so all the circle roles together with the rolling circle we have this sphere of random dots which you see moving it is clear that far from the point of contact far away the rotation is faster and at that point of contact you see the axis of rotation so there velocity is 0 and the point of the curve is connected with that center of the rotation and the tangent is orthogonal to that so every word as set for the plane works on the sphere and the same is true for for other
constructions so you don't really
need a mechanical construction so for the ellipse for example
we have a point and a circle around it and another point inside the circle and we join a moving point on the circle to this other point and we take this symmetry line between those 2 points and intersected with the radius and then of course this isn't ISA sale triangle because it has the symmetry line and therefore the sum of these 2 distances is the radius of the circle so clearly this as a construction of the units with its a geometric what gap are you just gonna construction do you say that in english I don't know I mean I've been told that English Gabon's elliptic shape English garden beds they have been drawn in this way now
again this works on the sphere no
presidents which so I don't explain too much I think you can see how I handle the menus and this is a
sphere now not represented by an random dots but by spherical ellipses and the construction of these ellipses is exactly the same as I explained in the training so you watch this and then I say something the the so again we have a circle With the midpoint and another point inside the circle we drop radius from a moving point on the circle we connect with this other point we take the symmetry plane in space between those 2 points the moving point and this other point and intersected with the sphere this is this great circle we call this great circles the symmetry line on the sphere between the 2 points and then the intersection with the radius has again the property that we have an ISA sealed triangle here and therefore the sum of these distances is the radius of that circle so if you notice I in change any word used exactly the same words as for the plane now I think my time doesn't permit me to do more planar curves but I would like to show the at least that's a tool examples of space curves and this I think I would like you to see without
the glasses for a moment without the glasses for a moment I think this picture out in spite of the fact that we have these undercuts here is not really interpreted in a spatial way however if you do an
eclipse it jumps into a three-dimensional object immediately and that's even better when you edit that is why the thing is called a torus knot is sitting on this torus the the and the dogs speak enough should I change them contain now when you move this it's even more three-dimensional and also because these out some 16 thousand pairs of green and red dots it's an impressive illustration of what our I system can do by identifying the correct pairs of points instantly to put them correctly into a the spatial image which you see and now I have a question to ask you of
course the have birds flying and like
that I is so much on the side so that this kind of
spatial impression cannot be generated so the idea is that the birds are able to synchronize subsequent pictures and I would like you to watch this and my question to you is for whom is it better to watch this was the 1 I and who from 1 doesn't make no difference so look at this for a moment with 1 eye the and was 2 eyes and then tell me what you see that just for my personal curiosity thing you could do that phase so for whom is it better with 1 are here minority OK my
my private tests lead to the same result I had preferred another outcome with you have to stick to the facts OK and then the other thing
which we are proud of of SIFT space curves of constant curvature and because of the short time I want only to a slightly more complicated 1 we look in use and how do we judge that this thing has constant curvature we run a circle which at each point optimality that means point ancient 1st and 2nd derivative at the same as for the curve and this circles obviously called the curvature circle off this curve and you can easily see that that circle has constant radius so therefore this is a space curve of constant curvature and what you can also see that now the thing rotates rather quickly and that portion of the curve looks a bit like a helix and now it rotates very little and that portion of the curve looks like a circle so circle like behavior what's left to start again it this so now the circle like behavior and now the helicoid like behavior and those the circle and the helicoid they are the only well space curves of constant curvature but of course of the harry Kautz doesn't close up at all and the fact that you can sort of combine these shapes to get close space curves I think it's very nice FIL now I think I start at 6 minutes past of the 1st 20 minutes all over OK uh that leaves me time to also to implicit curves and the so this is this thing now I don't think that tells you very much you have looked at the equation may be and another equation that each of those 2 equations determines the surface and if you draw all this picture together with the surfaces then you understand much better on how an implicit curve is made it is the intersection of these 2 space groups and since I have used these pictures in my courses this particular picture can also explain how would you do extreme on under the side conditions and them so it's a graph of the function and a vertical cylinder the vertical cylinder describes the site conditions and the maximum in of the function inside that cylinder is the maximum on this curve the so that's why we use this and you can also see an advantage of this dotted the rendering of the surfaces they can rotate and never obscure the curve very much I mean their positions will almost everything is projected in the same direction then you don't see much but most of the time you can clearly see the curve that you want to see and roughly see the surfaces so these are these uh and um then the they are other kinds
of curves which are also found
so for example Hilbert's cube filling curves is made from such an initial peace and then the next iteration scales this piece bound by factor 2 and put states copies together no this more complicated thing also starts at this point and ends at this point so again you can scale it down by a factor of 1 half and put a copies together that looks like this same thing again and this is the limit of what the screen will tool but of course is still 5 way from the limit of the Hilbert sets so this now 1 example without the glasses please the the so this Snowflake Curve I guess is well known I won't say much and and do a slight change we take an extra version of this and what action did is I he started with a fairly simple tessellations and then made the boundaries more complicated and what you see here in the 1st step is every trying edge has been replaced by or green red blue is know this we repeat every edge is replaced by a similar zigzag and again and again and again and say well at the limit of the screen it looks a little bit like the limit but Of course not really and now this is exact can be more honest pointed of course and then you get a family of curves which starts from the triangle and end up by a space-filling curve 1 can view this with fewer iterations I can set the parameters of this location and I will do very little here I would just take tool please easier to use the mouse when you're sitting and standing OK and now it's doing the same thing but only very small iteration steps and you see how the zigzag is just getting and then eventually the things close up 2 equilateral triangles and the same happens all the time and so the space this the fitting curve is sort of made up or its subprime approximations are made up of such curves running around trying to so much for the poorest the
now as you can see when you
watch a sequence what would collection of math lapses most people find surfaces more beautiful than them a curves
and I'm no exception to that but as I said whoops I'm missed what I wanted to show the that as such it looks like a C shown and near challenged maybe not so surprising but what I find surprising is that this thing is really made out of circles so this is where the glasses please and you can see a pair of circles and the related parameter lines connecting them run across via daughter snail shells so I think this could should convince you that what you see is a one-parameter family a 1 parameter family of circles which gives a very good impression of this natural object the the OK those events that 1 you kill this kind and what
else that I want to show now we
have a large collection of minimal surfaces but it's not very easy to get OK I can stop right away like no OK this that I will show 1 mole movie instead the so all this 1 now
this 1 is going to the and there's a lot of to see now you have a catenoid with the yellow inside now the heavy colored now of the knowledge with the yellow outside again the harry colored again the catenoid now with the helicoid it with the with the the color inside and so on and if you look at the parameter rectangles then you can see that their size and angles doesn't change so what you see is that the information which really can be made by very thin metal and you can more effect metal cheat in the way but this illustration shows you and I didn't shoulders and an active because it's difficult to to kind of the 2 sides to in an actress in a clear way so that you see this distinction between 1 and the other side of the surface and almost any minimal surface allows for long story and um we have this to give them with our program we have this documentation and about every object we have a description and you see formulas how the object I made and for the minimalist minimal surfaces we have long explanations which 1 can read but nobody ever wrote to us that he actually did to read it and finally the financial I 2 Implicit surface again
1 illustration of what the I can do and 1 illustration of what we can do it's much slower than the rate tracing of the stuff on unfortunately so I can rotate this thing but it is still in an active and what you don't see when you take off your glasses so that the mixture of red and green in all those yellow pick such changes but the I can take that apart and make to really 3 D picture for you out of these shades of yellow and green of red and green and I think I have time to run 1 more the in that
once again going to you think and forth the so I think maybe you want to know how these functions are made that forgot to say that n together with the this we have an action a new entry which says flow to the minimum set that can be done with rate traces but can be done with these dots and sold the surface which you have seen is sort of a thickening of this curve and all the other implicit surfaces of
high genus which we show I made in this way that some combination of curves and their distance from these curves as far OK thank you very much FIL and about the but you can use my group the play was wondering about the
thanks the some there is the king of the rule the I do not know at all so I you going to say that I just want to sit at the 4th talk is constant so the person could not come on so we're ready for questions at the back of the book
and you would and use the so
you're you distributes about the random dot patterns on the surfaces but that's along the track and you can they have they have
an equally distributed on the
surfaces and that distribution is good enough for critique of the policy plan to do to use those in numerical computations of eigenfunctions of the class and on its surface so we spend a lot of time getting those equidistributed on those of us the only public you with the
dogs that bias because you that there could be a into a sort of internally how do you move the dogs that fast is set it's about 20 thousand starts so which require some computational power is it's just hot coded in C or a lot of what are you doing and
surprisingly I mean this is a very old machine this morning 10 years old and even the predecessors were able to do that and we move the camera we don't move the DOS and then in the company there these machines are fast enough to do that I mean my at tiring still running and 1 of the things I like to do especially the mentally completely this all the time from the little from 2 to 1 billion by just holding the difference in the memory that humanist that we now these machines have so much faster it's unintelligible and it won't be long and then you can also with our what so what was the movie you rotate and I have a question myself and we we have having this whole workshop with the you know a different way of and projecting a
presenting the 3 D objects in space and on was just running looking at your and your objects some do think if there is a great too easy a connected to things to connect this program is the many animations and the geometries that you have insight that Poland's because then you could really you know kind of touch them and see them in front of you will in a new room you moving developed the analysis you have you have seen in the
implicit space craft example How to service intersect and their works sort of reasonably well but when you get into morning at school when the surfaces intersect pretty tangentially and then uh but the resolution power of the the the screen of the idea I don't know isn't good enough uh so that you interpolate those points into surfaces now when you have machines which had 10 times faster than this 1 you can of course use 10 times more darts and also I can make these dots Lu disks and maybe you would
like to see that in all but but
so it's not the disk is you know me I miss the so so this is what the disks of really fairly large size look like and they had just tangential disks at the
points would of those random dots and since this is a service which is given by a
function we know its gradient so we know the normal at each point so it's trivial to draw the tangential disk that these points and even this thing you can rotate well on this machine not too fast but still the I mean because you want to look at it the moment but this location is still fast enough for demonstration purposes so this is not a ray trace the digit woman yeah you think about the did you also think about and optimize the distribution of the point so in the not at a random point but um in a
way and such that the curvature so is the wild like pre-B behind in
some way of course we had these connections to the partial differential equation people we optimized for it for distribution and then of course what you see is what and when you just see the random dots they emphasize the contour by themselves you don't have to do anything because they are more points in the project indirect and um if you look at this from the side I mean I still learned
how to draw upon was my hand when I was a student and a but you don't really have to work to see the Conte was that they can automatically I have experiences with trying to to use dots which I get to pick
the edges of the screen that doesn't work to some extent but anything regular with you do produces artifacts that you don't want to see I mean I learned that already in the eighties when we 1st computed minimal surfaces there were always around the branch points of the coordinates and they were trouble and we had holes in the pictures and what any you who had not made the picture look and the 1st question they asked what others holes so in the same way I had in the 1st lecture we also heard don't produce artifacts they will just 2 people immensely and therefore we do not use regular
the OK thank you thank you again the