What's New in 3D-XplorMath

Version 10.9 March 2015

 

 

NOTES: The 3D-XplorMath webserver is located in the Mathematics Department of The University of California at Irvine.

The 3D-XplorMath website resides on this server and has the URL : http://3D-XplorMath.org,. A gallery of 3D-XplorMath created mathematical exhibits, called The Virtual Math Museum, has its home on the same server at the URL : http://VirtualMathMuseum.org.

Please report any anomalous behavior (aka "bugs") in 3D-XplorMath to the developers:  Hermann Karcher <unm416@uni-bonn.de>  and Richard Palais <palais@uci.edu>. We are also happy to hear your suggestions for improvements and additions to the program.

 Professor David Eck of Hobart and William Smith Colleges has created a cross-platform Java program called 3D-XplorMath-J. We expect that will eventualy have all the features of the Pascal-based Macintosh-only version of 3D-XplorMath. The current version is available at: http://3d-XplorMath.org/j/index.html

Since version 10.7, 3D-XplorMath runs native on Intel Macs as well as on PowerPC based Macs. The difficult work of porting the earlier PowerPC only source code to make this possible was carried out by Adriaan van Os. For further details see below.

Changes to 3D-XplorMath between versions 10.8.1 and 10.9

Work on the conversion to Linux and Windows is still in progress.

The Plane Curve Category:
All moving decorations of the curves can also be run and saved as movies - so that they can be sent to people asking questions. Via the Action Menu osculating circles can be added to implicit curves in two ways. A demo is added that shows osculating circles to involutes. The parametrized curves can be shown together with the graphs of their component functions. For completeness the rendering of the curves as graphs in R^3 was also added.

The Space Curve Category:
All moving decorations of the curves can also be run and saved as movies.

The Polyhedron Category:
The presentation of the dual polyhedra and their relation to the original polyhedra has been improved.

The ODE Categories:
Computing speed on the one hand and CPU-independent delays on the other hand have been improved.

The Conformal Maps Category:
An explanation of Stereographic Projection was added, accessible via the Action Menu. And an Erase-entry for the segments and circles which can be added to the grid.

The Surface Category:
A new Action Menu entry allows to search for closed geodesics on implicit surfaces, because 3D-XplorMath chooses as initial point of the geodesic a symmetry point of the surface, and many closed geodesics through this point exist. The sensitivity of the mouse is not yet documented. The feature should be tried out on the implicit torus and the implicit ellipsoid. - Parametrized Surfaces of Constant Width have been added. They can be rotated inside a surrounding cube.

The Fractals & Chaos Category:
On the Henon Attractor one can now find periodic points. The period 2 and period 4 orbits are added by default. We did not find period 3 and period 5 orbits. Orbits of periods between 6 and about 18 can be found. The Henon map expands so much that for larger periods the program does not find starting values for the algorithm that converges to an orbit of the chosen period.

 

Changes to 3D-XplorMath between versions 10.8 and 10.8.1

While version 10.8.1 started out as a bug-fix revision of Version 10.8, we have taken the opportunity to make some more substantial changes. In particular:

Starting with version 10.8.1, it is no longer required to run 3D-XplorMath from within the 3D-XplorMath ƒ folder. This is because the application file is now a bundle that contains the folder 3DFSDocs within it. (Previously, 3DFSDocs was a subfolder of the 3D-XplorMath ƒ folder and the application looked for it there when the user selected "About This Object..." from the documentation menu.) It is still possible to access the 3DFSDocs; just "right click" (i.e., click while holding down Control) the application file and select "Show Package Contents" from the resulting drop-down context-dependent menu.

We have been given financial support by the Hausdorff Institute to convert our present MacIntosh-only code to Linux and Windows. To make this conversion proceed more smoothly, we have done extensive code cleaning operations.

Changes to 3D-XplorMath between versions 10.7 and 10.8

As always, we have improved the documentation (even though no one seems to read it :-). We would very much like to know the extent to which our program is being used on "old" systems and hardware. The CodeWarrior compilation runs on OS 10.6.8 and earlier, the new Free Pascal compilation runs on OS 10.4.11 and later. Both compilations are combined in the Universal build that we distribute. Since our point clouds have been used for Monte Carlo type numerical computations, we have switched to the Mersenne Twister, a state of the art random number generator. - The control of the playback speed for movies was too hidden, we added a slider control (visible only while a movie is playing).

The Plane Curve Category:
A demo showing rotating gear wheels has been added. The teeth of the gear wheels have circle involute flanks.

The Surface Category:
We have added more "decorations", in particular to the Implicit Surfaces: Families of Curvature lines, polar geodesic grids and parallel geodesic grids which can be moved with the mouse over the surface. - Surfaces such as the Boy Surface are difficult to grasp as a whole; the first attempt to make a demo of revolving Meridian-Möbius-Bands resulted in Möbius bands hopping around in space. We have developped a new morph-with-background where the whole surface is rendered as point cloud and the Meridian Bands are now visibly moving on the surface. This led us to improved morphs for various other surfaces (Cross Cap, Right Conoid, Bianchi-Pinkall). - We experimented with rendering Implicit Surfaces by putting tangential disks at the points of a point cloud. These renderings are fast enough for mouse rotation, but since we cannot take these disks too small, the quality is still less than that of a flat shaded parametrized surface.

The Fractals & Chaos Category:
The Henon Attractor can now be rendered in Hit-Count-Mode. This gives some impression of the invariant density on this complicated set.

Changes to 3D-XplorMath between versions 10.6.1 and 10.7

With the departure of the Apple PowerPC to Intel emulator, Rosetta, version 10.6.1 and earlier of 3D-XplorMath could not run under Apple's OS 10.7 (Lion). We are grateful to Adriaan Van Os for converting our CodeWarrior source code to be compatible with FPC, the FreePascal Compiler. The FPC-compiled Intel code runs native on Macintosh Intel-based machines and in particular works under Lion.  Many bugs had to be fixed, some were found by Adriaan, others during the extensive testing that the converted code required. Some of the improvements are not visible, for example the coordinates of the random points on implicit surfaces are computed with much higher precision since they have been used for numerical computations, e.g., of eigenfunctions of the Laplacian. It is possible to read in point cloud data that represent surfaces and we have written code to reconstruct approximate normals directly from the point cloud.

The Space Curve Category:
For the user-defined implicit space curves there is a new Action Menu entry that displays as dot clouds the two surfaces whose intersection defines the current implicit space curve. To illustrate how this feature can be used we added buttons to select three default examples. In one case, one of the surfaces is the graph of a function, and the other is a vertical cylinder; this illustrates how the determination of extrema on subsets works.

The Surface Category:
We improved some default morphs, adding multiple defaults in some cases (e.g., the Klein Bottle and Boys Surface). The main change is that we have started to add "decorations" to the images of surfaces. As in the earlier additions to other categories they are reached by entries in the Action Menu. For parametrized and for implicit surfaces one can add the principal curvature fields, move (with the mouse) the pair of principal curvature circles along the surface, and also move the family of normal curvature circles at a point. (Note that these normal circles form naturally ocurring families of cross-caps.)  One can also add geodesics to the surfaces. For implicit surfaces,  the mouse position gives the initial point and, as long as the mouse is pressed, the initial direction changes with the movement of the mouse. For parametrized surfaces a geodesic spray can be added, its center moving with the mouse.

Other Categories:
No visible changes.

Changes to 3D-XplorMath between versions 10.6 and 10.6.1

One major reason for this release is to make available additions to the documentation. There are now ATO (About This Object) texts for all objects in the following categories: Plane Curves, Space Curves, Conformal Maps, Parametric Surfaces and Fractals. We also corrected a serious omission: one documentation folder was empty by accident in the previous version and this has now been fixed. There are a number of improvements to the program features that involve the anaglyph images: in particular Phong shading was extended to anaglyph, and the anaglyph button now handles all anaglyph cases. The behavior of the 3D-axes was also improved. Switching to dot cloud rendering is no longer handled in the Action menu; it is now treated in parallel with wire frame and patch style in the View Menu. For a few objects the program switches automatically to dot cloud rendering but returns to the previous style if a new object is selected. Since the size of pixels on modern monitors has decreased, there are now more situations were pressing a number key will switch to dots of that size.

The ODE Category:
The Forced Duffing Oscillator (1D, 2nd order) has been improved in that the orbit computation and the Poincare map are fully synchronized. One can run the Poincare map and then toggle to orbit computation, which superimposes the orbit over the Poincare image. See the ATO for details. For two-dimensional first order ODEs, one can now switch between different numerical methods in the Action Menu. Also a demo-mode has beens added, illustrating in a visually self-explanatory way three simple numerical methods.( The Runge-Kutta demo does still needs verbal explanations.)

Other Categories:
No changes.

 

Changes to 3D-XplorMath between versions 10.5.6 and 10.6

 

The additon of a click-image interface to operate touch-screens (see below) introduced some bugs (and exposed some hidden old ones). All known such errors have been corrected. For the convenience of touch-screen users without a keyboard, we added a Pause button, a button to toggle between anaglyph stereo and monocular viewing, and a button to start default morphs. These features are also convenient on a computer monitor, in particular when used in lectures. Please report any misbehavior of the pause- and abort-functions. Only one piece of contents is added, in

The Surface Category:
For ray-traced, implicit surfaces we have added an anaglyph version. Note that the yellow of the image is the mixture of the red for the left eye and the green for the right eye; the intensity changes of the two component colors are not really visible in the composite yellow, but when viewed through the red-green anaglyph glasses our eyes get enough information to create a 3D impression better than what we had hoped for---quite amazing.

 

Changes to 3D-XplorMath between versions 10.5.5 and 10.5.6

 

The documentation for all of the mathematical objects (the ATOs) is now available from the download page, independent of the program itself, as a single cross-platform ZIP file. We have also created five Quicktime movies (about 10 minutes each) that explain how to get started and how to use 3D-XplorMath. These 'streaming videos' can also be obtained from the download page.

The Click-Image Interface:
The biggest change, however, is the addition of a second user interface that we call the "click-image" interface. It was designed for possible use with a touch screen in mind (and we have tested it on a touch screen in the Oberwolfach Museum). One may toggle between the new and old user interfaces using the second entry of the 3D-XplorMath Menu. In the click-image interface the Category Menu is replaced by a page of twelve Category Icons. If one of these is clicked then the corresponding page of Object Icons appears and one can choose an object by clicking its icon. In this way one does not need to know the names of objects. Once touch-screens become more widely available, it should be fun to select the objects in that way.

The Plane Curve Category:
All mechanically created curves have improved default morphs. The morphs include now the drawing mechanism and the tangent construction as the curve (i.e. the drawing pen) varies. Archimedes' angle trisection is available in the Action Menu of the Circle. Strophoids are added because they have the same drawing mechanism as the Cissoid - We have added 'Show Caustics' in the Action Menu of most curves, one can vary the angle of the rays against the curve with a small slider.

The Space Curve Category:
An implicitly defined curve is the intersection of two (implicit) surfaces. In some cases this implies that the implicit curve is the singular curve of the boundary of the intersection of two solid objects. We emphasize this intersection (so far: see Two Cylinders). For tubes and pairs of strips we have added the possibility to omit the less informative parameter lines. - When 'Parallel Frame' is checked and 'Show Repere Mobile' is chosen in the Action Menu, then we show not only the frame but also the curvature vector and its history. In the last entry, 'Show Frenet Integration', we reconstruct the curve from its parametrized curvature vector (given in a parallel frame).

The Surface Category:
Small improvements for the rotation by mouse were added, in particular, by pressing TAB one can (again) rotate surfaces in rough Patch Mode. The raytracing of Implicit Surfaces is now much faster, with only a small loss in perfection.

The Polyhedron Category:
All the Archimedean solids are now available, mostly from the Action Menu by different kinds of truncations (including the Snub Polyhedra). The remaining two Archimedean solids are obtained as 'modified' standard truncations of the Cubeoctahedron and the Icosidodecahedron. (Recall that in this Category the default morphs depend on the selection that is made in the Action Menu.) Since objects with Platonic symmetry have been known since 2500 BC, in the form of stone ball ensembles, we have added in the Action Menu "Show as Stone Balls". The balls are represented by random dots in Wire Frame and as fine subtriangulation of the Buckyball in Patch Mode. These subtriangulations are available for many other cases after "Create Subdivided" is selected; they are shown as triangulations of a sphere. - All truncated polyhedra can be shown inside the not truncated one. In particular the snub truncation can thus be better understood.

The Fractals & Chaos Category:
As a twin to the Feigenbaum exhibit, a "UserDefined Feigenbaum" has been added since this kind of display is a convenient way to locate periodic attractors in Newton iterations. The curves that represent attractors in the Feigenbaum Display extend to what looks like "curves through the chaotic portions of the tree". This can be looked at more closely with two new entries in the Action Menu. One shows 1000 iterations in the usual vertical line, then the next 1000 iterations in the neighboring column of pixels and so on for 400 columns. The result illustrates a density distribution belonging to the particular iteration chosen. Therefore we also determine by counting in pixel size intervals the density function of this distribution. - The Dragon can tile the plane in several ways, we have added two of them, see the Action Menu. - For various choices of parameters the Henon system gets periodic attractors. These can more easily be looked for with the new entry 'Continue Mouse Point Iteration' because this allows to continue with much higher accuracy than is available for mouse selected iteration points.

Other Categories:
No changes.

 

Changes to 3D-XplorMath between versions 10.5.4 and 10.5.5

 

We encourage users to look into the documentation, almost every object now has an explanation text "About This Object (ATO)". In addition, the folder "UserDocs" contains material that we used in lectures - as an example how to use this feature.
The file menu entry "Save Settings..." that allows to save the current program state and later open 3DXM in exactly this state, has been updated and extensively checked.
More rare bugs were eliminated.

The Space Curve Category:
A new display of curves, "Show as Pair of Strips", has been added in the Action Menu. It is similar to "Show As Tube", but while the tubes mainly emphasize the curve as 3D object the strips emphasize the curvature properties of the curve.
We have added V.Jones' braid list so that the first 249 prime knots can be viewed in Jones' braid representation. New braid words can be entered in a dialogue. The braids can either be displayed circular and almost planar, or on the surface of a cylinder. A second addition allows to modify the trefoil knot into a sequence of prime knots. The default Lissajous curve is now another prime knot.

The Surface Category:
The conic sections entry "Planes, Cones and Spheres" has been expanded into two views. The first shows the cone intersected by a plane, the default morph changes the inclination of the plane. The second view shows the Dandelin Spheres, the default morph keeps the intersection curve the same and varies the cone angle (down to a cylinder).
All surfaces can now be rendered as point clouds, including the multi-tile surfaces, see "Hopf Fibered Linked Tori".
The contours of all parametrized surfaces can be displayed. To study the contours choose wire frame or point cloud rendering. In patch display the contours are only useful if in "Light Sources" one has chosen "Ambient Only" - for line drawings combined with the painters algorithm to suppress invisible parts of the surface. Pressing Left/Right Arrow during computation changes the line width for the contour.
We have expanded the Dirac Belt demo: 3-frames can be added to the belt to indicate the family of motions and a second morph displays only the first half of the belt thus showing Feynman's Plate Trick.

The Polyhedron Category:
The exhibits have been improved with a very young audience in mind. Dotted hidden edges can be added from the Action Menu. The entry "Show Relation to Cube" now works also for the Rhombic Dodecahedron and the pyramids on the faces of the cube can be flipped into the cube by pressing Left/Right Arrow.

The ODE Category:
The Forced Duffing Oscillator has been added (1D, 2nd order) together with submenu entries that offer playgrounds for experimentation. This includes the Poincare Map. See the ATO.

The Fractals & Chaos Category:
To the list of C-values we added two more with linearly neutral attractors, but large attractor bassins (see Between Attractors). The Action Menu entry for "Julia Sets": 'Show C-value in Mandelbrot Set' also shows a rough images of the Julia sets as the mouse moves.

Other Categories:
No changes.

 

 

 

Changes to 3D-XplorMath between versions 10.5.3 and 10.5.4

 

 

 

The background choices "white" and "custom" were added to wire frame drawings of Anaglyph images. Various small errors are fixed that were connected with switching decorations or with morphing. The occasional but old bug in Phong shading is gone.

The Plane Curve Category:
A decoration was added to the Cassini curves. It shows that the product of the distances to the foci is constant.

The Space Curve Category:
When Parallel Frame is selected we show the normal curvature vector and draw the normal curvature curve.

The Conformal Maps Category:
In many cases the image grid, when viewed on the Riemann Sphere, left a hole around infinity. This hole is closed.

The Surface Category:
Surfaces can now be saved as triangulated surfaces and such data can be read and rendered.

The Fractals & Chaos Category:
The Fractal Curves can be mapped (from the Action Menu) with the conformal maps z^2 and exp(z). This emphasizes complex functions as Conformal Maps and it leads to a better understanding of selfsimilarity. Before mapping a curve, the origin can be chosen with the mouse (except for the Julia sets, these are mapped 2-1 onto themselves with z^2 - c). One can select from the Action Menu to show the first component of any of the Fractal Curves z(t) as graph [t,Re(z(t))], thus providing a large collection of continuous, non-differentiable functions. Note, that for c-values from the boundary of the Mandelbrot set a continuous parametrization of the Julia set (of z^2 - c) cannot work in a stable way since totally disconnected Julia sets are arbitrarily close. For the boundary c-values from our list more than the first 20 backwards iterations are computed and ordered correctly.

Other Categories:
No changes.

 

 

 

 

Changes to 3D-XplorMath between versions 10.5.2 and 10.5.3

 

 

 

 

Old and new bugs were found and fixed and some imperfections when switching between viewing modes were corrected.
The need to adapt the program to projectors of different power and to darker or brighter room light is presently met as follows: If a number key is pressed when Create is selected from the Action Menu, then the dot-size used for drawing is set to that number. (This works for Planar Curves, Space Curves, Surfaces and Polyhedra).

The Space Curve Category:
Those closed space curves of constant curvature that have normal line symmetries (dd=0) can now be morphed as closed curves via a new entry in the Animate Menu. The two parameters aa, bb are automatically adjusted to user-selected changes of cc, ee. More interesting and more robust is another new morph that automatically adjusts bb to keep the normal symmetry lines intersecting in one point. One can watch the morph until a new closed curve appears and then stop the deformation, preferably when the Action "Add Symmetry Elements" is selected.

The Surface Category:
All parametrized surfaces can now be rendered as Point Clouds. Point cloud representations of surfaces are becoming popular because they also arise from laser scans of physical objects. Note that the number of points in a Parametric Surface Point Cloud is set in the "Set Resolution and Scale..." dialog (accessed via the Settings menu).
New triply periodic minimal surfaces (from the list of A. Schoen) have been added. Since these surfaces look very different depending on how their fundamental pieces are assembled, a corresponding choice is now available; one assembly presents one side of the surface as "outside" and the other assembly presents the other side as "outside". (For surfaces that carry straight lines, the two sides look the same so this choice is omitted).
Some numerical imperfections of Scherk With Handle have been removed.
The feature that "an interrupted patchmode drawing gets completed in wireframe" now also works for minimal surfaces.
The Paraboloid and the Ellipsoid can be shown with focal rays and their reflections (i.e., parabolic antennas and whispering galleries). One may morph the light source away from the focus.
There is a submenu of the Surface menu labeled "Voxel Clouds". This is for a new (and experimental) subcategory, still under development, designed for visualizing volume densities. The current examples include a constant density and the low quantum number Hydrogen atom orbitals.

The Polyhedron Category:
The remaining Archimedean Polyhedra, namely the edge-truncated regular polyhedra, have been added. The Icosahedron can now be morphed inside the Octahedron. The computation of all morphs has been made flicker-free.
The Cube has a new Action: "Show Intersection with Plane": a non-moving dotted plane is added and the mouse-controlled cube removes those dots that are inside the cube (this works in wireframe only).
The Icosahedron has a new Action: "Add Borromean Link". In wireframe, three pairs of opposite edges are completed to rectangles that are linked in the Borromean fashion. (This image morphs in an interesting way !)

Other Categories:
No changes.

Changes to 3D-XplorMath between versions 10.5.0 and 10.5.2

Two bugs, in the Lattice Models and in the Planar Curves were fixed.
The parameter DotSize in the Settings Menu can be used to change linewidth of curves and of parameter lines of surfaces.
The View Menu got another entry: 'Anaglyph Objects Into Room'. This results in the same anaglyph renderngs but the objects are placed distinctly in front of the screen.
The program still runs fine under OS 9.2: some computations are faster, but the rendering of rotating 3D objects is much slower.

The Plane Curve Category:
The cubic curves were given more demos to illustrate the geometric addition on these curves.
Two color morphs were added.
The osculating circles can smoothly be drawn up to inflection point tangents. Also, normals are drawn in the correct direction, even if their end points are well outside the 'graphics sphere'.
The ODE-defined curves now allow the same Actions as the other curves.

The Space Curve Category:
Those closed space curves of constant curvature that have reflection symmetries (in normal planes) can now be morphed as closed curves via a new entry in the Animate Menu.
The dotted sphere on which the spherical ellipses are shown is now dotted with orthogonal families of spherical ellipses having the same focal points. A second default morph changes the focal distance.
A new Action, 'Show Frenet Integration' has been added. It computes for all space curves the principal curvature vector with respect to a parallel normal frame. This two-dimensional curve is the angular velocity vector of the time dependent parallel frame. This curvature-curve, the resulting rotational motion of the parallel frame and the integration of the first frame vector to the given curve are shown.

The Surface Category:
A new entry 'Diracs Belt Trick' has been added. It opens in anaglyph stereo with a cyclic morph that is interesting because it shows a non-obvious property of the topology of the three dimensional rotation group.
The Bianchi Pinkall Tori has a second default morph. It shows---in stereographic projection---the continuous rotation of these tori around one of their Hopf fibres in S^3. In R^3 this is a continuous conformal deformation that turns the torus inside out (one torus of the family has to pass through infinity). Some tori are mapped onto themselves when turned inside out: they have an antiinvolution with a circle as fixed point set showing that they are rhombic tori.

Other Categories:
No changes.

Changes to 3D-XplorMath between versions 10.4.2 and 10.5.0

Some user interface imperfections and rarely ocurring bugs were fixed.

In the dialog resulting from selecting 'Eye Separation' in the Settings Menu one can now adjust another view parameter called 'Dot Size'; it changes the line width for surfaces, planar and space curves.

The Plane Curve Category:
A Curves of Constant Width exhibit has been added. We also added more involved decorations to some curves, in particular to the Deltoid. The mechanical construction of the Lemniscate was made compatible with its default morph.

The Space Curve Category:
We have added more examples of closed constant curvature curves. We added 'Satellite Knots' in the Action Menu of the Trefoil, Figure Eight and Granny Knots, and also the Spherical Ellipse, an un-knot. The satellite knots indicate how one can approximate any space curve by a curve of constant curvature.
The main addition is an involved demo from rigid body kinematics and dynamics. The most easily understood motion is in the Action Menu for 'Spherical Cycloids' under 'Show Rolling Circle'. Next, all space curves can be used to give an 'Angular Velocity Function' in the observer space, with the corresponding action showing the associated motion. Next, all space curves can be used to give the 'Angular Velocity Function' in the body space and again the corresponding action shows the associated motion. Finally, a new space curve, 'Eulers Polhode' has been added; it is the solution of Euler's first order ODE and, when taken as 'Angular Velocity Function', the associated motion is the physical motion of a free rigid body. Parameters aa, bb, cc are the principal moments of inertia, and parameters dd, ee, ff are the components of angular momentum with respect to the moving frame. The picture of the polhode shows both, the angular momentum curve and the angular velocity curve. Both are intersections of quadratic surfaces and these are also shown.

The Surface Category:
The surface family x^p + y^p + z^p = 1 that joins the cube to an octahedron was added. For some implicit surfaces we added an Action 'Flow to the Minimum', to make it clearer how the defining function is constructed.
After creating a surface, it can now be saved for use with Mathematica either as a .m file or as a Mathematica Notebook (.nb file).

The Conformal Maps Category:
The Conformal Maps have two new features: 1.) Circles or Intervals, that are added to the domain using the mouse, stay with the map when its view is changed (scaling, Gaussian plane to Riemann sphere, anaglyph stereo). 2.) One can choose a tangential approximation of the map that follows the mouse pointer. This emphasizes conformality and helps to understand branch points.

The Polyhedron Category:
No changes.

The ODE Category:
No changes.

The Fractals & Chaos Category:
The Sierpinski Curve was added to the Fractals. Its default morph joins it to an equilateral triangle.

The Sound Category:
No changes.

Changes to 3D-XplorMath between versions 10.4.1 and 10.4.2

The documentation describing the various exhibits, the so-called ATOs (About This Object), is now more than twice as large as the code of the program. Most ATOs contain suggestions for experimenting with the selected object.

Here are some of the new features added to various categories:

The old 'Save Settings / Open Settings' feature was not work reliably for some time, but is working again. Old settings can still be read by the newest version of the program, however one should not read new settings with an old version of the program version since new control variables have been added.

The Plane Curve Category:
To the mechanically generated curves we have added a moving plane, i.e. a square of random dots that is rigidly connected to the drawing mechanism. This makes evident an important concept, the instantanous center of rotation.

The Space Curve Category:
The spherical rolling curves have a dotted sphere rigidly attached to the rolling wheel to illustrate the momentary motion as in the planar case. The 3D-impression in monocular view is stronger if, following David Eck, the curve is drawn as 'Thick Curve' (select from the Action Menu).-- We have found closed constant curvature curves that are knotted, or even have nonvanishing torsion (select parameter values from the Action Menu).

The Surface Category:
The minimal surface subcategory now has a User Defined entry for the Weierstrass representation. -- The Whitney Umbrella and the Right Conoid now have default morphs that emphasize their pinch point singularity (see ATO). -- A new entry, Snail Surfaces, produces realistic pictures from simple formulas.-- A mouse related bug in the Sine-Gordon surfaces has been cured. -- The Hopf Tori and the Pinkall Tori have two families of conformal deformations: cc moves the center of the stereographic projection, ff rotates around a Hopf fibre on the torus. -- A two sided User Coloration has been added, with a default tuned for Hopf Tori.

The Conformal Maps Category:
The conformal map image of z --> (z + cc)/(1 + conj(cc)z) shows the unit circle and the parametrized image. This allows demos of hyperbolic motions.

The Polyhedron Category:
The drawing with gaps has been expanded to the truncated, stellated and nested polyhedra. David Eck's draw as 'Thick Curve' also works in these cases. -- For Tetrahedron, Cube and Icosahedron we have added 'Show Relation with Octahedron' and 'Show Relation with Cube' has an instructive morph. -- One can now choose in the Action Menu: 'Show Anaglyph Demo'. It explains how the 3D-impression of anaglyph pictures comes about. -- One can also choose: 'Show Central Projection To Sphere'.

The ODE Category:
For ODE(3D) 2nd Order we have prepared for all three cases of charged particles in a magnetic field sets of initial conditions that illustrate the behaviour of different types of trajectories (available in the Action Menu).

The Fractals & Chaos Category:
All Fractal Curves now have a default morph in which the Hausdorff dimension increases from 1 to 2. See the Action Menu for more variations. -- On the Julia sets one can mark the periodic points of period 3, 4 and 5 (Action Menu) and this information is used to improve the computation of certain Julia sets (Action Menu - C Values - Between Attractors). Additional special C-values from the boundary of the Mandelbrot set have been added. -- A User Defined planar map iteration has been added, together with Action Menu entries to allow experimentation with it.

The Sound Category:
This is a new category that has just been added. The so-called Shepard Tones (aka "The Ever-Rising Note") is the first exhibit in this category.

 

Changes to 3D-XplorMath between versions 10.4 and 10.4.1

We have made a considerable number of small bug fixes and improvements to the user interface. In particular, there are numerous speed-ups included in this version.

The Plane Curve Category:
Now most planar curves come with a "decoration" that explains how the curve is defined. The latest addition is a mechanical construction of the Lemniscate. We have also made the various decorations perform in a more uniform fashion---for example, now all decorations remain visible if one stops their animation with a mouse click.

The Space Curve Category:
We have added Curves of Constant Curvature to the exhibits (with many closed ones in the default morph) and curves of constant torsion (again closed ones in the default morph). We had not seen closed space curves of constant curvature treated before and found them interesting to look at. Although computed from their Frenet differential equation, all of the entries for explicitly parametrized curves in the Action Menu remainavailable for this new exhibit, and the same holds for Curves of Constant Torsion. See the ATOs for further interesting details.

The Surface Category:
In the minimal surface subcategory there is a new Action Menu item: Show Associated Grids. It shows on the left the grid on which we perform the numerical integration, in other words, this grid defines which parameter lines appear on the surface. On the right we either show the Gauss image of the integration grid (left), or, for surfacesof genus > 0, we show the image grid under the complex third coordinate function.

The Fractals & Chaos Category:
Instead of showing only still pictures of the Henon attractor and of the Feigenbaum tree we have added some dynamics so that one can now see how the final pictures evolve from early approximations. The use of colors shows clearly where a mixing behaviour occurs. Also, in the Henon Attractor, if you hold down Command, you can drag out a "zoom rectangle" on the screen (i.e., the window will zoom to a magnified view of the part of the attractor included in that rectangle). Important improvements have been made to the Julia Set animations.

Changes between versions 10.3 and 10.4

There were many changes to 3D-XplorMath in moving to version 10.4.. Some that lie beneath the surface may not be obvious to a casual user, such as improved coding and the removal of a number of subtle (and a few not-so-subtle) bugs. But there are also numerous quite visible changes that we hope our users will find both interesting and beautiful. These include the addition of many new objects, new kinds of rendering methods and animations, and improved documentation. We will discuss these below, category by category, but first, here is a quick description of several of the more noteworthy new features.

Phong Shading Option. 3D-XplorMath has always used so-called flat shading to color surfaces in patch mode. This means that for each patch (always rectangular in 3DXM) the correct color is calculated from the normal at the center of the patch, and the entire patch is given that color. This is quite fast, but unless the grid-spacing is very small it leads to unaesthetic sharp color changes at patch boundaries. A much more accurate and smooth color rendition is obtained by first calculating the normals at the vertices of a patch, and then using barycentric coordinates to interpolate the correct normal at every interior pixel and using this interpolated normal to choose the color of the pixel. This method, called Phong shading , is much more computationally intensive, and the computer graphics texts of ten years ago advised that it should be reserved for off-line use when very high quality was required, since it was too slow for real-time computer graphics. But the last decade has seen an enormous rise in the popularity of computer games that for their full appreciation require very high bandwidth graphics and, for competitive reasons, this has pushed computer manufacturers and graphics card fabricators to make very substantial improvements in the performance of the graphics sub-systems of even relatively modest computers. About six months ago we decided to see if this made it possible to do Phong shading to show surfaces in patch mode in 3DXM. We were very happily surprised at the quality of the resulting images, and even more at the speed of rendering. Running on even a moderately fast machine it is quite suitable for real-time use, and on a fast dual G5 the speed difference between Phong and flat shading is barely perceptible. The program still starts up using flat shading, but the user can shift to using Phong shading (and back to flat) using the View menu.

Dual Image Stereo Modes. 3D-XplorMath has always used the anaglyph technique to create stereo images of 3D objects. This means that the left-eye and the right-eye images are rendered on the monitor in different colors and are superimposed. They are then seen separately by the two eyes through the use of a different colored filters over each eye. This is remarkably effective and also has the advantage of being very inexpensive, since the bi-colored glasses required are both cheap and easily available. The major drawback to the anaglyph technique is that the color filters preclude having high quality color rendition. Over a hundred years ago, before anaglyph stereo, it was already common to take photographs of the same scene from slightly different viewpoints and then view these with a so-called stereopticon. That is, the left and right eye images are placed side by side and lenses or prisms used to focus the two images appropriately on the left and right retina. Of course this method works just as well in color as in black-and-white. In mathematical visualization, the objects do not have any intrinsic color, so anaglyph stereo is fine for most purposes, but it is still desireable to have the option of showing full-color stereo images of 3D objects, and we have now implemented this. In fact there are two so-called "dual-image" stereo modes now in 3DXM. The View menu, in addition to the former "Monocular Vision" and "Anaglyph Stereo Vision" items also has two new items, namely "Cross-Eyed StereoVision" and "Parallel-View Stereo Vision". In the first, the left-eye view and right-eye view are widely separated, with the left-eye view on the right and right-eye view on the left. With a little practice, most people can learn to fuse the two images by crossing their eyes (but don't do it for too long; it causes eye strain). In parallel view stero mode the images are closer together and not reversed, and must be viewed with some sort of stereopticon. Suitable inexpensive stereopticons can be found at several places on the web (for example, here ).

The 3D-XplorMath Web-Site and Gallery    The 3D-XplorMath project has had its own Website for nearly ten years. Originally it was a place from which one could download the latest version, and later a modest Gallery was added where various images created by the program were on display. Recently, the Gallery has been given a very major face-lift and upgrade by the new 3DXM Webmaster, Xah Lee, and it has now been expanded to a complete gallery of images of all surfaces in the 3DXM Surface menu. Each image can be rotated with the mouse, and has explanatory documentation, and many of them have accompanying animations in the form of Quicktime movies. Over time, we expect to expand this to include the other 3DXM categories (plane and space curves, polyhedra, conformal maps, ODE, lattice models, waves, fractals) so that gradually it will become a museum and explanatorium of mathematics.

The Plane Curve Category

The Plane Curve menu has been rearranged into more logical groupings.

The User Graph item has a new feature that shows approximations to the graph using Taylor series, Lagrange polynomials, and Fourier series.

It is now possible to choose to have tick marks on the x and y axes, using the final item on the View menu

There is a new animation mode for plane curves that we call Color Morphing. When this is selected, instead of a series of curves being created on different canvases that are "played back" as a motion picture, all the curves are drawn on the the same canvas but in different colors. In effect, color replaces time to distinguish the different stages of the morph. This can be very striking and show features that are not easily evident in the animated version since it allows careful comparisons between the various stages. We recommend trying this on the Cassinian Ovals.

There is now much improved descriptions of the various plane curves. Xah Lee has imported much of the material from his Famous Curves website into the ATOs for a number of the plane curves. As an example, select Astroid from the Plane Curve menu and then select About This Object from the Documentation menu.

A new Epi- and Hypocycloids selection has been added to the Plane Curves menu.

The default morphs for many of the plane curves have been improved.

The constructions that 3DXM shows (automatically) for Ellipses, the Epi-and Hypocyloids can now be compared with analogous constructions on the sphere associated to two new items in the Space Curves category, Spherical Ellipse and Spherical Cycloid. See below.

The Space Curve Category

While this category has been relatively unchanged for a long time, in Version 10.3 there are a number of significant and interesting changes.

The Space Curves menu has a new group of Spherical Curves, i.e., curves that lie on the Sphere. Two of these, the Spherical Ellipse and the Spherical Cycloid are close analogs of the planar ellipse and planar cycloid, and their demos are conceived with the goal of emphasizing the close analogies between Euclidean and Spherical Geometry. When the Spherical Ellipse is the selected object, choose Show Spherical Ellipse Demo from the Action menu, and when the Spherical Cycloid is selected, choose Show Rolling Circle.With both, be sure to select About This Object and About Spherical Curves from the Documentation menu. Show Osculating Circles with Evolutes is pretty and interesting for the Spherical curves and several of the other space curves as well. The Stereo View enhances this section a lot.

The family of torus knots is no longer alone: we have added the connected sums of two torus knots, placed on a genus 2 surface.

The Surface Category

As usual, perhaps the most striking and important changes have been in the Surface category. In particular there are now two new rendering modes, one for parametric surfaces (Phong Shading---see the detailed discussion above) and one for implicit surfaces that we call Dot Cloud Rendering. For the latter, we have developed an algorithm that we believe is new (although the mathematical idea behind it is very old!) to sprinkle dots randomly on an implicit surface with a density that makes the number of dots in any region of the surface proportional to its area. This works particularly well in stereo where it gives a method for seeing all sheets of a complex immersed surface at once and detecting the structure of the self-intersections. It also displays the contours of the surface well. One can also use this to do interesting vision experiments with the dot-clouds in the dual stereo modes mentioned above.

There are two new surface coloring options, color by Gauss curvature and color by mean curvature. These are available only for parametric surfaces, and are turned on from the Surface Coloration submenu of the Action menu:

   Action>Surface Coloration>Hue = Gauss Curvature   and   Action>Surface Coloration>Hue = Mean Curvature 

There have been very significant additions and improvements to the ATOs of many of the surfaces.

In the minimal surface subcategory, additional dihedral symmetries have been added to DoubleEnneper and Symmetric4Noids, and "wavy" perturbations have been added to the ends of PlanarEnneper and Riemann.

Twisted Scherk now allows deformations almost to the degenerate limits.

An entry Show Normals, has been added to the Action menu. This can be used to illustrate that the Gauss map of a minimal surface does not depend on the associate family parameter.

A new sub-category called Surfaces of Revolution has been added. It displays surfaces of revolution having constant curvature in various different senses. Be sure to check out the CMC (constant mean curvature ) case---namely the so-called Unduloid, and its About This Object.

Chuu-lian Terng has programmed in a remarkable collection of surfaces (the Ward Solitons) that show graphs of energy density as a function of time for a class of two-dimensional solitons. The animations associated with these surfaces are striking---and a little mysterious too. Since they are rather sophisticated and not easy to explain in a few words, we recommend choosing one of them and then selecting About Ward Solitons from the Documentation menu.

With the help of Paul Bourke and Luc Benard, we have made it possible to export surfaces created in 3D-XplorMath directly (using the File menu) as either .inc files (readable by POVRay) or as .obj files (readable by Bryce and other 3D programs).

The Conformal Map Category

No changes.

The Polyhedron Category

For each of the regular polyhedra (except for the cube itself) there is now an item "Show Relation to Cube" at the bottom of the Action menu. Selecting this brings up a graphic that shows the polyhedron incscribed in or circumscribed about a cube in a way that suggests how the polyhedron can in fact be created from the cube.

The ODE Categories

Only the Lattice Model subcategory has been changed significantly. We have in particular gone over the numerical algorithms with considerable care since we have plans to do a careful rerun of the famous Fermi-Pasta-Ulam experiments of half a century ago that both ushered in the use of computers as an experimental tool in mathematics and theoretical physics, and also led indirectly to the discovery of solitons. There is one new viewing mode that is connected to these plans: during the display of a lattice model evolution in transverse mode, if the Shift key is depressed then the display will shift to a graph of the energy distribution among the various modes as a function of time (corresponding to the graph on page 12 of the research report on the FPU experiments).

The Wave Category

No significant changes.

 

The Fractals & Chaos Category

The accuracy of computations of Julia sets has been improved.