User's Manual
This is a reference describing how each menu item and tool
in Group Explorer works. It is arranged to be referenced, and not
intended as a tutorial. If you're interested in tutorials, see
the Tutorials page. You can also return
to browsing all of our documentation.
Table of Contents
To jump to a topic in this page, click on it in this Table of Contents.
File | Open group
When the user clicks this menu item, the current group is closed and a standard Windows "Open" dialog box
appears, from which the user can browse
and choose a .gp file for Group Explorer to open.
After the user selects a group to open, Group Explorer will display a
Cayley diagram for the group (provided one exists), and open any of the
four standard windows that the user had open last time they viewed a
group. Those four standard windows are the Navigator, the
Multiplication Table, the History View, and the Permutation View.
In the case where the user selects an invalid group file, Group
Explorer will not open the file but will instead report an appropriate
error message. This may happen if the file has been corrupted, or if
a non-.gp file was opened, or if the user is developing their
own group and has not done so correctly. Such error should not occur
with any .gp file that comes with Group Explorer, but in the
event that such an error does occur, please help by reporting it
immediately.
Users who wish to develop their own groups should refer to
the group Authoring page.
File | Close group
When the user clicks this menu item, all child windows of the main Group Explorer window disappear,
as well as any explorations the user has done in them.
File | Print current view...
The contents of the active child window can be printed using this menu
item. They will be printed close to the top left corner of the
printed page, roughly the same size they appear on the screen.
When the user clicks this menu item, Group Explorer displays a standard
Windows "Print" dialog box, for choosing preferences like the number of copies,
the printer settings,
etc. To cancel printing, the user can click the Cancel button on
that print dialog. After setting options as needed, clicking OK proceeds with printing.
For more information on printing, refer to the tutorial called
Saving and Printing Images.
File | Save current view...
The image contained in the active child window can be saved to a
.bmp file using this menu item.
When the user clicks this menu item, Group Explorer will display the standard
Windows "Save" dialog box,
by which the user can choose the filename into which to save the image.
To cancel the save, the user can click the Cancel button on that save
dialog. After selecting a file, clicking OK proceeds with the save.
The saved file can later be opened in any application that opens .bmp files, e.g. MS Paint.
For more information on printing, refer to the tutorial called
Saving and Printing Images.
File | Save VRML world...
The 3D scene shown in the active child window can be saved to a VRML
.wrl file using this menu item.
When the user clicks this menu item, Group Explorer will display the standard
Windows "Save" dialog box,
by which the user can choose the filename into which to save the VRML
world. To cancel the save, the user can click the Cancel button on
that save dialog. After selecting a file, clicking OK proceeds with
the save.
The saved file can later be opened in a VRML viewer, or a web browser
that has a VRML plug-in, like the
Cortona
plug-in for most browsers.
For more information on printing, refer to the tutorial called
Saving and Printing Images.
File | Exit
Clicking this menu item closes Group Explorer immediately.
The Group Menu is only enabled (available to be used) when Group Explorer
has a group open. When the Group Menu is grayed out
and cannot be used, the user must first open a group, using File |
Open group.
Each .gp file contains a considerable amount of
information about the group it describes, and Group Explorer computes
still more information as it loads the group. All the information
Group Explorer knows about the open group can be read in a text
presentation in the Group Information window. Clicking the Group |
Information menu item opens that window, and the user can use the X in the
upper-right-hand corner of the Group Information window to close it.
Here is a sample image of a Group Information window containing
information about the symmetric group S3.
The text in the Group Information window is largely self-explanatory,
and so here follows merely a short explanation of the text visible in the
above image.
 | The group information begins with the name of the group, its other
names, and the group author, as declared in the .gp file.
The "Embedded in:" line specifies which permutation group was used to
embed the group in question. Group Explorer requires group authors
to embed each group inside a group of permutations Sn
for some integer n > 1. In this case, S3
is embedded in itself. |
| Each group can specify several sets of generators, so the "Loaded 3
ways to generate S_3" section lists each set of generators, and labels
the active one (see the Group | Change Generators menu item below).
In this case, there happen to be three in the .gp file, so
that's how many Group Explorer loaded. |
| Similarly, each group may have several representations for its
elements, and these are listed in the "Loaded 2 ways to represent S_3"
section. The sequences of integers (like 1 2 0 under
Representation #1) are the internal representation of the elements of
the group; this is how Group Explorer sees them. The end user need
not be concerned with this, but it is sometimes useful information to
those who are writing new groups. |
| The "Loaded 1 object of symmetry for this group" section describes
the one object of symmetry that was described in the .gp
file, and that explanation is fairly self-explanatory. Were there
any custom Cayley diagrams defined for the group, they would have
appeared in a similar section immediately above. |
Group | Reset
When the user clicks this menu item, the group returns to the state it was in when
it was opened. Specifically, it navigates immediately to the identity element of the group
in every open view,
and erases the contents of the History View.
Group | Change Representation >
When the mouse cursor hovers over this menu item, a
submenu listing one or more different representations for the open group
appears. The user can click on an option from that list to change how
Group Explorer represents group elements.
For example, groups often contain the representation option
"Permutations as cycles," meaning that the name of each element in the group will be
the cycle notation for the permutation that the group author used to
encode the group element. Another representation
option is a table of representations which may have been
explicitly created by the group author, or generated by Group Explorer from the group author's
names for the generators. Such options will
appear in the form "Table #1 (e.g. a, b)," where the "a" and "b"
are the names of the group's generators in that representation scheme,
shown on the menu as an example.
Group | Change Generators >
When the mouse cursor hovers over this menu item, a submenu
listing one or more different vectors of generators for the open group appears.
The user clicks on an option from that list to change what generators the Navigator
provides for navigating
the group. Each of these options is of the form
< a, b >, where "a" and "b" are the names of the generators. (For groups
generated by one element, the lists are of the form < a >,
and so on for other dimensions of groups. But a group need not
always be generated by the same number of elements.)
Warnings
Note that changing generators often causes warning messages to appear.
There are two reasons why such warning messages may pop up.
- If a window displaying an object of symmetry is open and that object
of symmetry cannot respond
to one of the newly selected generators, Group Explorer will
give a warning to this effect, and then close the object of symmetry.
- If a window displaying a Cayley diagram is open and the diagram's arrows do not
include all of the
newly selected generators, Group Explorer will give a warning to this
effect. The problem in such a situation is that navigation in the Cayley diagram
using those generators may not always follow
the diagram's arrows, and may therefore be confusing. Group Explorer will not close Cayley diagram
windows in this situation.
Group | Create Cayley diagram >
When the mouse cursor hovers over this menu item, a submenu listing all available Cayley diagrams for the
open group appears. Clicking on one of these options opens
a new window in Group Explorer displaying the diagram. Group Explorer will
always have at least two Cayley diagrams available for any group with
fewer than 5 generators. For most groups there are about five basic
options, each of which can be further customized using
the Edit Cayley Diagram dialog box.
If the group has five or more generators, the Create Cayley diagram menu
item will be grayed out and unusable.
In
addition to diagrams that Group Explorer can generate, the author of the group may have specified additional
custom Cayley diagrams for the group. The presence of custom Cayley
diagrams usually indicates that they are more aesthetically pleasing than
the auto-generated diagrams that are named only by the shapes they assume
("Rectangular," "Cylindrical," etc.). But custom Cayley
diagrams have the drawback of being less editable; see
the tutorial on editing Cayley diagrams.
The user can click and drag
on Cayley diagram windows to rotate the
diagram in three dimensions. A popup menu for changing various viewing options
is available by right-clicking on the diagram. See below, under
Cayley Diagrams for further discussion of
diagram options.
Cayley diagrams and the Navigator
Newly opened diagrams highlight the group element previously navigated
to using the Navigator, and on display in the Permutation View and History
View, and highlighted in the Multiplication Table. Thus all views in
Group Explorer stay synchronized. The highlighting in a Cayley
diagram (a grey circle behind the
highlighted element's node) follows navigation as long as the diagram
remains open.
Group navigation is temporarily disabled while any object of symmetry
or Cayley diagram is still responding to a previous navigation (i.e. still
animating). The speed of such animation is an option that is
available in the popup menu on both Cayley diagrams and objects of
symmetry.
Group | Create Symmetry object >
This menu item will only be active if the open group
has at least one symmetry object defined. The author of the
group may not have included any symmetry objects, in which case this menu
item will be disabled (greyed out and unusable).
When the mouse cursor hovers over this menu item, a submenu listing all
available symmetry objects for the open group appears. Clicking on
an option from that submenu opens a new child window displaying the object.
Symmetry objects and the Navigator
Symmetry objects respond to navigation within the group (using
the Navigator) by rotating in place,
according the group author's specifications. When the mouse cursor
hovers over a generator in the Navigator (or the user presses its shortcut
key), the symmetry object draws a dotted arrow and an axis of rotation indicating how
the object will respond to navigation using that generator. Using the generator (by clicking on it in the Navigator, or by pressing its
shortcut key while holding Shift) initiates an animated response
on the part of the symmetry object, which rotates itself in accordance
with the
previewing.
Group navigation is temporarily disabled while any object of symmetry
or Cayley diagram is still responding to a previous navigation (i.e.
still slowly rotating). The speed of such animation is an option
that is available in the popup menu on both Cayley diagrams and objects of
symmetry; bring up that menu by right-clicking inside the window.
For other options on that popup menu, see
Symmetry Objects below.
Window | Hide/Show Navigator
This menu item reads "Hide Navigator" if the Navigator is visible, and
"Show Navigator" if it is invisible. The Navigator window can also
be hidden by clicking the X in its upper right corner. Unlike Cayley
diagrams and objects of symmetry, there can be only one Navigator window.
Whether the Navigator is hidden or visible, navigating the group is
still possible using the keyboard commands the Navigator defines.
For more information on the Navigator, see the section below entitled
The Navigator.
Window | Hide/Show Multiplication Table
This menu item reads "Hide Multiplication Table" if the Multiplication
Table window is visible, and "Show Multiplication Table" if it is
invisible. The Multiplication Table window can also be hidden by
clicking the X in its upper right corner. Unlike Cayley diagrams and
objects of symmetry, there can be only one Multiplication Table window.
Whether the Multiplication Table window is hidden or visible, it still
responds to any navigation operations the user performed, so that it
remains synchronized with all other windows, even when invisible.
For more information on the Multiplication Table, see the section below
entitled The Multiplication Table.
Window | Hide/Show Permutation View
This menu item reads "Hide Permutation View" if the Permutation View
window is visible, and "Show Permutation View" if it is invisible.
The Permutation View window can also be hidden by clicking the X in its
upper right corner. Unlike Cayley diagrams and objects of symmetry,
there can be only one Permutation View.
Whether the Permutation View is hidden or visible, it still responds to
any navigation operations the user performed, so that it remains
synchronized with all other windows, even when invisible. For more
information on the Permutation View, see the section below entitled
The Permutation View.
Window | Hide/Show History View
This menu item reads "Hide History View" if the History View window is
visible, and "Show History View" if it is invisible. The History
View window can also be hidden by clicking the X in its upper right
corner. Unlike Cayley diagrams and objects of symmetry, there can be
only one History View.
Whether the History View is hidden or visible, it still responds to any
navigation operations the user performed, so that it remains synchronized
with all other windows, even when invisible. For more information on
the History View, see the section below entitled
The History View.
Window | Tile
Arranges all visible, non-minimized child windows in a grid such that each
child is of equal size.
Window | Cascade
Places the first child window in the upper-left corner of the main
Group Explorer window (without changing its size) and then lays all other
windows in order in front of it, moving each new window slightly down and
to the right of the previous, as if laying out cards for a magic trick.
Window | Arrange All
Arranges any minimized child windows to be in the lower-left of the
main window, in a row.
Help | About...
Displays information about where Group Explorer was conceived and
built, and by whom.
The Navigator serves three basic but essential purposes in Group
Explorer.
- It displays the current set of generators,
- allows the user to navigate the group using those generators by clicking or pressing
shortcut keys,
- and gives the user access to changing the shortcut keys
associated with each generator.
Seeing the group's generators
In the Navigator shown above, one generator is displayed, and it is
called "a." The name of the generator always appears after the word
"Generator" in the Navigator view. If the user had asked
Group Explorer to represent group elements using permutation cycle notation,
the line would read "A = Generator (0
1 2)" instead.
The information communicated by the "A = " portion of the line is that
the key "A" is the shortcut for "Generator a." It is no coincidence
that "A" is the shortcut for "a"--the Navigator tries to use a letter that
appears in the generator's representation, so that the shortcut keys make
sense.
From the Group menu, the user can change
which set of generators should appear in the Navigator, and can change the
representation of whatever generators are already there. The results
of such actions show up immediately in the Navigator.
Using the displayed generators
Via the mouse:
When the mouse hovers over the phrase "A = Generator a," several
things occur:
| The text itself becomes highlighted, indicating that Group Explorer
recognizes the user's interest in that generator. |
| Any visible Cayley diagram uses an array of translucent yellow
circles to show where the highlighted generator would navigate within
the Cayley diagram. |
| Any visible symmetry object draws a dotted arrow to indicate what
effect using that generator would have on the object of symmetry. |
If the user clicks the mouse on the phrase "A = Generator a," several
things occur:
| The Permutation View changes its
contents to reflect the action of the generator
clicked. If the Permutation view was displaying an
element x, and the user clicked on the generator y, then
Group Explorer performs the operation of left-multiplication by y,
arriving at the group element yx. |
| The History View appends text to its
contents describing the navigation that took place. |
| The Multiplication Table
highlights the element to which the action navigated, to coincide with
the History and Permutation Views. |
| Any visible symmetry object will begin to rotate, as per the preview
shown when the mouse hovered over the generator. |
| Any visible Cayley diagram will move its gray highlighting circle
according to the navigation that was previewed when the mouse hovered
over the generator. |
Via the keyboard:
The user can also preview the use of a generator using the keyboard.
This is accomplished by
pressing and holding the shortcut key for that generator. To use a generator,
the user can press the generator's
shortcut key while holding Shift on the keyboard.
Why does the Navigator sometimes not respond?
The Navigator is only listening to user input if Group Explorer is not already
in the process of manipulating some of its views according to previous
commands. That is, after invoking a generator so that Group Explorer
is animating some of the currently-visible windows, the user must wait
until those views finish animating before navigation becomes available again.
Changing the shortcut keys for generators
Clicking the "Change Shortcut Keys" button in the upper-right
corner of the Navigator brings up a new window, shown here.
In this simple example, the user has three options.
| Click on the drop-down list to choose a different letter or number
that should be the shortcut for Generator a. |
| Click OK to close the window and make the user's changes take
effect. |
| Click Cancel to close the window and revert to the old shortcut
keys, ignoring any changes made in the "Change Navigator Keys" popup
window. |
If a group has more generators, the window will have a drop-down list
for each generator. However, at no point can the user assign
two different generators the same shortcut key. Group Explorer will
report an error message and switch the values back to their former
(unique) setting if the user attempts it.
What the Permutation View shows
When a group is first opened, the Permutation View will look much like
the one above, with the identity map displayed on the right. This
indicates
that the group element currently receiving focus is the identity element.
Navigating through the group causes the map depicted in colored circles to
change to represent the new group element to which the user navigated.
The Permutation View will always depict the same group element that is
highlighted in any visible Cayley diagram,
which will also be the same element that has affected any visible
object of symmetry.
Why the Permutation View?
Cayley's theorem, a central theorem of group theory, says that every
group is isomorphic to a group of permutations. This is useful for Group Explorer,
because Group Explorer only needs to be able to speak the language of permutations,
and it can therefore be taught the structure of any group. Thus Group Explorer might
rightly be called "Permutation Explorer," because those who author groups
are required to encode their groups as groups of
permutations.
The Permutation view has the following basic uses.
| It provides a visual view of any group, with no effort on the part of
the group author. |
| It provides the user hands-on interaction with permutations
and their composition. |
| It provides the user first-hand experience of how groups fit in
permutation groups. |
What the History View shows
When a group is first opened, the History View shows the word
"Start" in its upper left corner, followed by the representation of the
identity element (in this case "e") indented on the next line.
The History View keeps a record of the user's navigations through the
group.
The History View shown above indicates that the user
navigated in the following way.
|
The user started at the identity element, as Group
Explorer forces one to do. |
|
The user then applied the generator r and it moved
them (naturally) to the element r, because using r is
the same as multiplying the current location (the identity) on the left by
r. |
|
The user then applied the generator r again and it
moved them to the element r. |
|
The user then applied the generator f, and it
moved them to the element rf. One might expect that f
applied to rr would yield frr, and indeed it does, but the
representation Group Explorer gives is the most concise one for an
element, and frr = rf. |
|
And so on. |
The Multiplication Table displays a complete list of the group's
elements as both column and row headings, and shows completely the group
operation within the table itself. In group theory, multiplication
tables sometimes go by the name "Cayley tables" or "group tables."
Many beginning abstract algebra textbooks contain a mention of Cayley
tables in their introduction (e.g.,
Fraleigh), but you can find information about them online at
PlanetMath
or
MathWorld. For this reason, the meanings and uses of
multiplication tables are not discussed in this manual, but rather only
Group Explorer's commands for manipulating them.
Highlight current element
The checkbox labeled "Highlight current element" toggles on or off the
dark black highlighting visible surrounding the identity element's row and
column in the above image. The black highlighting will move as the
user navigates the group, to stay synchronized with all other views (Cayley
diagrams, objects of symmetry,
History View,
Permutation View). Unchecking
the checkbox removes the highlighting, yet the Multiplication Table still
tracks where the highlighting would be were it present, so that even if
navigation takes place while the highlighting is off, it can be restored
still synchronized with the other views.
Large font
The image above shows the Multiplication Table with a small font used
to write the element names in the grid. For a larger font (and hence
a larger grid), the user can check the "Large font" checkbox. For
smaller group tables, it is usually preferable to have a large font, and
vice versa.
Move column left/right
If the user clicks a square within the Multiplication Table, its column
header will darken gray to indicate that the user has selected the given
column. The user can then move that column left or right within the
multiplication table by clicking the "Move column left" or "Move column
right" buttons, respectively. Group Explorer retains the same order
of the elements as row headings and column headings, so moving a column
left also necessarily moves the corresponding row upwards, and similarly
for moving right and downwards. This feature is present so that the
user can organize the multiplication table into any order.
Edit table...
The "Edit table..." button brings up a dialog box through which the
user can tweak an extensive list of options which Group Explorer respects
when generating the Multiplication Table. The Edit Table dialog box
is discussed thoroughly in its own Tutorial.
What is a Cayley diagram?
A Cayley diagram is a directed graph showing the relationships among
generators of a group. In the above polar Cayley diagram for the
dihedral group D4, there are two generators, one (r)
drawn as red arrows and one (f)drawn as blue lines. The eight
spheres in the diagram represent the eight elements of the group, and each
element is labeled with its name.
A red arrow from one element to another in the diagram indicates that
left-multiplication by r takes the source element to the target
element. That is, we see a red arrow from fr to f in
the diagram, indicating that r times fr equals f in
D4. Similarly, the blue line represents the
generator f, and so connecting rrf and rr with a blue
line indicates that f times rrf equals rr. The
blue lines need no arrowheads, because f is an order-two element,
and so is its own inverse.
The current element
The identity element in the Cayley diagram shown above is highlighted
with a gray circle behind it. This highlighting identifies the
"current element" in the diagram, and will be the same element that is
singled out in the Multiplication
Table, the Permutation View, the
History View, and any other visible Cayley
diagram. It will also be the element that is represented by any
visible object of symmetry.
Interacting with Cayley Diagrams
There are three ways to interact with a Cayley diagram.
- Navigating through the group moves the current element highlighter
within a Cayley diagram.
- The user can spin any Cayley diagram in space as a three-dimensional
object, to view it from whatever angle they choose. This is
accomplished by clicking on the diagram with the left-mouse button and
dragging to spin the object.
- The user can right-click on the diagram to bring up a popup menu
with several options that affect how the diagram is drawn. There
follows below a list of each option on that popup menu, and a
description of its use and purpose.
Label placement >
This option has four possible settings.
| Front - Labels always appear in front of everything else in
the diagram, as if they were laid on top of the scene by a transparency,
only two-dimensionally near the spheres they label. Labels are
never obscured in this situation, but often obscure parts of the
diagram. |
| Back - Labels behave just as with front, but as if the
transparency were behind the image. Labels never obscure parts of
the diagram in this situation, but are often obscured by the diagram. |
| Throughout - Labels float in the three dimensional world of
the diagram, and thus may be in front of some parts of the diagram and
behind others. |
| None - No labels are present in the diagram. |
Labels translucent
This option has two settings, on or off. Labels are translucent
when a checkmark appears next to this option, meaning that one can
partially see through the labels to the objects that lie behind them.
When this option is off, labels are completely opaque.
Arrowhead style >
This option has three possible settings.
| Caret - When arrowheads are drawn on lines, they are thin and
shaped like a caret (^). |
| Wedge - When arrowheads are drawn on lines, they are a bit
thicker than a caret, shaped more like an actual bow-and-arrow
arrowhead. |
| Block - When arrowheads are draw on lines, they are shaped
like a triangle for maximum thickness and presence. |
Arrowhead placement >
This option has three possible settings.
| Center - Arrowheads drawn on lines will appear half way from
its start to its end. This option often seems most natural to make
the arrowheads easiest to spot. |
| Two-thirds - Arrowheads drawn on lines will appear two-thirds
of the way from its start to its end. On some diagrams, if all
arrowheads were centered, many of the arrowheads would collide as lines
crossed each other. Thus placing them a bit off-center with this
option is helpful for uncluttering the diagram. |
| Endpoint - Arrowheads are drawn on lines at the point where
the line ends. This sometimes makes them less difficult to spot,
but is a common place to put arrowheads. |
Animation rate >
This option has six possible settings.
| Very Slow - When the highlighting circle in the diagram moves
from one element to the next, it does so excruciatingly slowly. |
| Slow - When the highlighting circle in the diagram moves from
one element to the next, it does so slowly. |
| Normal - When the highlighting circle in the diagram moves
from one element to the next, it does so at an average pace. |
| Fast - When the highlighting circle in the diagram moves from
one element to the next, it does so quickly. |
| Very Fast - When the highlighting circle in the diagram moves
from one element to the next, it does so almost immediately. |
| Immediate - When the highlighting circle in the diagram moves
from one element to the next, it simply disappears from the old location
and appears at the new location, with no animation delay. |
For more complicated diagrams, it is useful to have a slower animation
rate so it is easier to see what is going on. With very simple
diagrams, the user often would rather not wait for the animation to
complete in order to continue navigating, and so a faster animation rate
is acceptable.
Brightness >
This option has three possible settings.
| Dark - The brightness of each color in the diagram is halved
so that darker, richer colors are used. |
| Medium - The diagram is displayed in the colors with which it
was defined, with no alterations taking place. |
| Light - The brightness of each color in the diagram is
averaged with pure white so that pastel versions of each color are used. |
Fog level (depth cueing) >
Depth-cueing is the use of a fog-like effect in a three-dimensional
scene to accentuate the distance of objects from the viewer. The
scene is drawn with more distant objects slightly foggier, blending more
into the background, so that nearer objects are more crisp, and depth in
the scene is more obvious.
This option has three possible settings.
| Full - Fogs out more distant objects with a rather thick fog. |
| Half - Fogs out more distant objects with a rather thin fog. |
| None - Turns off depth-cueing. |
Sphere shading
This option has two settings, on or off. When sphere shading is
turned on, a check mark appears next to this menu item, indicating that
the nodes in the diagram will be drawn as spheres with a light side that
faces the light and a dark side that is hidden from it. When sphere
shading is turned off, the nodes in the diagram are a uniform color,
unshaded, and appearing more flat.
Set defaults to these
Clicking this menu item instructs Group Explorer to save the current
settings of the diagram, as made using the popup menu, so that all future
Cayley diagrams are created with those same settings as their initial
settings. This information is saved in the GroupExplorer.INI
file in the application's working directory.
Edit diagram...
The "Edit diagram..." button brings up a dialog box through which the
user can tweak an extensive list of options which Group Explorer respects
when generating the Cayley diagram. The Edit Diagram dialog box is
discussed thoroughly in its own Tutorial.
Symmetry and groups
Symmetry objects are named thus because they are three-dimensional
shapes whose symmetry is significant for their use in Group Explorer.
Above is shown a square, whose three-dimensional symmetries form a group.
The group they form is the dihedral group D4, and thus
the above symmetry object is part of that group. The generators of
D4 can be used to manipulate the square in all the ways
one could do it with one's hands, and by experimenting in this regard one
can convince oneself that D4 is indeed the symmetries of
the square.
The image above shows not just a square, but one whose four corners
have been colored four different colors, and also piercing the center of
the square is an axis about which the square might spin. A dotted
arrow is indicating one of the square's symmetries; the square could be
rotated 90 degrees counterclockwise about the axis piercing its center,
and it would end up in the same spatial position, except for color.
When this happens, the colored halos that surround each sphere in the
picture do not move, but rather remain in their original places so
it is clear to the observer the configuration in which the square began.
It is an interesting to play with objects of symmetry in the following
way. Open a group and one of its objects of symmetry, then begin
clicking generators at random until the object is in a quite different
arrangement, and it is not obvious how to return it to its original state.
Then, using only the Navigator and the symmetry object window, return it
to its original state. (Using a Cayley diagram is cheating because
it's a complete map of how to get back home!) For small objects of
symmetry this game is very easy; for the icosahedron (in the group A5),
it can take some careful thought and experimentation.
Interacting with Cayley Diagrams
There are two ways to interact with an object of symmetry.
- Navigating through the group manipulates the object of symmetry.
When considering a navigation (either by hovering the mouse over the
generator name in the Navigator or by pressing its shortcut key without
holding shift) the object of symmetry will preview the manipulation, as
in the image above.
- The user can right-click on the object to bring up a popup menu with
several options that affect how the diagram is drawn. There
follows below a list of each option on that popup menu, and a
description of its use and purpose.
Label nodes/homes
Each of these menu items ("Label nodes" and "Label homes") has two settings, on or off.
| When "label nodes" is turned on, a check mark appears next to this
menu item, indicating that the nodes (spheres) in the symmetry objects
will be labeled with a number below and to the left of them. |
| When "label homes" is turned on, a check mark appears next to this
menu item, indicating that the original locations of the nodes (spheres) in the object
will be labeled with numbers above and to the left of them. These
numbers will appear just above those that are caused by "label nodes." |
The default is to have these options
off; no such labels are shown in the symmetry object above.
The purpose of these options is to allow the user a fail-safe
method for seeing whether the object has returned to its original
configuration. That is, if the user is trying to use the group
generators to return the object to its starting configuration, it is
important to tell if each "node" has returned to its "home." Thus we
draw a unique number near each node, and a unique number near its home.
When they match, the object is in its "home" position.
A better way to do this is with the "Circle homes" option,
below, which uses colors. But in the event that the viewer has
difficulty distinguishing the colors, this option is a fall-back.
Circle homes
This option has two settings, on or off. When "Circle homes" is
turned on, a check mark appears next to this menu item, indicating that
the scene should be drawn with colored halos indicating the "home"
locations of each node. That is, each colored node in the scene came
from a certain original location when the window was first opened; that
location will be marked with a colored halo so that the user can tell
whether the object is in its original orientation. The window shown
above has homes circled.
Animation rate >
This option has six possible settings.
| Very Slow - When the object responds to a generator by
rotating in place, it does so excruciatingly slowly. |
| Slow - When the object responds to a generator by rotating in
place, it does so slowly. |
| Normal - When the object responds to a generator by rotating
in place, it does so at an average pace. |
| Fast - When the object responds to a generator by rotating in
place, it does so quickly. |
| Very Fast - When the object responds to a generator by
rotating in place, it does so almost immediately. |
| Immediate - When the object responds to a generator by
rotating in place, it does so with no intervening animation to cause delay. |
For more complicated objects, it is useful to have a slower animation
rate so it is easier to see what is going on. With very simple
objects, the user often would rather not wait for the animation to
complete in order to continue navigating, and so a faster animation rate
is acceptable.
Brightness >
This option has three possible settings.
| Dark - The brightness of each color in the object is halved
so that darker, richer colors are used. |
| Medium - The object is displayed in the colors with which it
was defined, with no alterations taking place. |
| Light - The brightness of each color in the object is
averaged with pure white so that pastel versions of each color are used. |
Fog level (depth cueing) >
Depth-cueing is the use of a fog-like effect in a three-dimensional
scene to accentuate the distance of objects from the viewer. The
scene is drawn with more distant objects slightly foggier, blending more
into the background, so that nearer objects are more crisp, and depth in
the scene is more obvious.
This option has three possible settings.
| Full - Fogs out more distant objects with a rather thick fog. |
| Half - Fogs out more distant objects with a rather thin fog. |
| None - Turns off depth-cueing. |
Sphere shading
This option has two settings, on or off. When sphere shading is
turned on, a check mark appears next to this menu item, indicating that
the nodes in the object will be drawn as spheres with a light side that
faces the light and a dark side that is hidden from it. When sphere
shading is turned off, the nodes in the object are a uniform color, unshaded, and appearing more flat.
Set defaults to these
Clicking this menu item instructs Group Explorer to save the current
settings of the symmetry object, as made using the popup menu, so that all future
objects of symmetry are created with those same settings as their initial
settings. This information is saved in the GroupExplorer.INI
file in the application's working directory.
|
