Slashdot Mirror
Qt vs MFC
Philippe Fremy writes: "I have just published and translated into English a comparison between Qt programming and MFC programming, which was written by Pascal Audoux (a fellow coworker). I am interested in feedback and would love to add quotes from developers that have used both toolkits." Here is the English version ("Qt vs MFC") as well as the French one ("Qt contre MFC").
A Linux user goes back.
/etc/fstab file so that it always automounted when plugged in. I was very impressed.
/dev/null, once I find where that actually is.
By Tony âoekNIGitsâ Collins.
Introduction...
In much of today's online news, we hear of how many people are migrating to GNU/Linux. What we don't seem to hear much of, is users going back to their old operating systems. The reason for this article is to say that I've done just that.
Yes, I've gone back. After three and a half years of trying to make GNU/Linux work on the desktop, I've decided that it's simply too hard for the average home user. Before I go into my reasons for going back, let me outline what I believe an 'average' home user is. Mr Joe Average is someone who wants to install their OS, boot it up, and it works. He wants to be able to upgrade his PC , and have the hardware work in a few short minutes. He wants to read email, browse the web, talk to his mates online, and play some games. Feel free to disagree with me, this is merely how I see myself. Note: I'm not referring to Grandma using Linux, or even my mum using it. I'm referring to average users who know a little about their computer.
Three and a half years; that's how long I've been trying to make Linux work on my desktop computer. Right about now, I'm sure that you are now screaming that I didn't try hard enough, or that I'm just plain stupid. Let me assure you that this is not the case. Stupid users don't doggedly stick at something for three and a half years, trying distribution after distribution in the hope of finding the holy grail of Linux desktops. They give up in less than a few hours of trying to (unsuccessfully) install RedHat Linux. Hear now my sad tale of why Linux isn't suitable for my desktop.
Some background...
The year is 1998. I've had my Windows '95 computer for around six months. Frustrated with the constant crashes, I desperately asked an online mate for help. Even though he was a windows user, he calmly suggested that I try something I'd never come across before...
âoeLinux, eh? Never heard of it.â
âoeOh, it's a free OS that you can download. Apparently it doesn't crash much. Just do an online search for it.â
Armed with this meagre knowledge, I set out on my quest for the ultimate stable operating system. I searched online, and found places where you could even buy copies of Linux! So, I left the comfort of my warm study, and returned forty minutes later with my first Linux boxed set â" RedHat Linux 5.2. After initially balking at the very basic installer (and few false starts), I had it up and running on my lovely AMD K6-233. I even got X working in no time at all. Then the system booted up for the first time.... and it was dead ugly. I had a very stable new OS, but I didn't even want to look at it. I was happy that I had several installed interfaces to choose from, but none of them appealed to me whatsoever. Wanting to download a nicer interface led me to my next problem.
I had absolutely no idea how to even get this nice, stable OS onto the internet! After reinstalling windows and RedHat in a dual-boot configuration, I got the help I needed by using Windows and USENET. Strangely enough, I can still remember the name of the long-suffering person who helped me get RedHat online, but that's another story. After looking around online, I discovered KDE. Only up to version one, it was the closest thing I had to a completely useable Linux system. I downloaded all the KDE packages for RedHat 5.2, only to discover another distro called Mandrake, that came with KDE preinstalled and configured. Back to my local distributor, and I was set.
Mandrake with KDE was exactly what I needed at that stage in my Linux using life, and I stuck with it for over a year and a half. Always seeking the 'perfect' desktop OS, I followed releases from version 5.3 all the way through to 7.0. Eventually I became dissatisfied with Mandrake, and briefly tried a number of other distros until I finally settled on Debian. I was impressed by the simple power, configurability, and the ease of upgrade that is apt-get. I felt good about being among the uber-elite Debian user community. Needless to say, I learned a lot about how to configure hardware under Linux during my time with Debian. I learned to sift through the old HOWTOs on Linux Doc until I found something suitable and accurate, I learned to utilize the power of USENET and IRC. Life was good.
Right now you must be wondering; âoeWhere is this leading? This guy seemed quite happy with Linux!â. True, I was. After a while, I decided I didn't want to have fine-grained control. I wanted something simple. I was getting tired of the 'stable' Debian release being so out of date, and the 'unstable' distribution being so... well... unstable. I got tired of having to recompile my kernel every time I got new hardware. I got tired of using command line to talk to my PC. It was time for a change. I had good experiences years ago with Mandrake, so I figured I'd try it again. As good as Mandrake 8.1 was, it wasn't what I was after. SuSE Linux 8.0 Professional (boxed set) was installed onto my PC instead.
I have to stop at this point, and say that SuSE Linux 8.0 (Pro) is the best Linux distribution that I've ever used. It has an easy installer, reasonable hardware support, and comes with the very good KDE 3.0. The box contains seven CDROMS, one DVD and three decent books that would help even the most inexperienced user get up and going. YaST2 is a decent graphical system configuration tool. When (not if) I go back to Linux, I'll definitely try SuSE again. However, there are quite a number of things that have improve (or change completely) before I'll consider going back. Read on for my brief list of things that must must get better before I'll switch back from the Microsoft camp.
Where GNU/Linux needs to improve...
X11
The X Window System is an awesomely powerful, network transparent graphical subsystem. It's perfectly suited to running applications from remote servers. However, this is NOT what a home user needs. My experience with X is that it's too big, bloated, slow and unstable to be any good to the home user. Most crashes that I ever experienced with Linux have been X's fault. My servers don't run X, and they never crash.
What home users need is something small and fast, so they can run local applications efficiently. I would like to see the X Window System dumped in favour of a hardware accelerated framebuffer, running something like directFB or Qtopia. Home users need a small, fast graphical subsystem, with built in 3d support. BeOS seemed to be on the right track before they went under.
Fonts are truly awful under X. Most distributions ship with appalling fonts, and there is no standard way to add additional (nicer) fonts to the system. Even after extra fonts have eventually been added, many applications (eg Abiword, Staroffice) refuse to use the new fonts anyway. Perhaps the framebuffer-based graphical subsystem I suggested could incorporate decent font support, and use a readable naming scheme as well.
Drivers
While having access to the latest version of the kernel is a good thing for developers, for home users it can be a nightmare. Got RedHat Linux 7.3? Perhaps you run SuSE 7.3 or Debian 2.2. You'll have to download a binary package specific to your distro. (I'm assuming that home users won't change their default kernel, but if they did, that binary package wouldn't even work!) Hardware manufacturers should be able to provide one single driver that works on all minor versions of a major kernel release. This way it would work will all current distros, instead of having to provide multiple binaries or source code. Hardware manufacturers don't want to give out the source, as this often gives away trade secrets about how their hardware is designed.
The solution seems to be to make binary drivers work on a variety of kernel versions. I'm not sure if this is even possible with the way the kernel is designed (I'm no kernel hacker), but it would go a long way toward making Linux more accessible to the home user. Even if the kernel needs to be redesigned to support this, then in my opinion, it should be done. Linux users are always clamouring for drivers... perhaps if the kernel had something like this, it might one day become a reality.
Hardware setup
While SuSE Linux 8.0 gave me some good experiences with hardware detection (such as automatic download of NVIDIA drivers), it also let me down as in this area.
The good: I recently borrowed a digital camera from a mate at work, to take photos of my case mod. Imagine how happy I was when I plugged it into my nearest USB port, and it was automatically configured (as a SCSI device) and mounted! SuSE even added it to my
The bad: Along came my new IDE CDRW drive. At AU$99, I couldn't pass up the purchase. Plugging it in gave me no joy. I was very disappointed that a device so common couldn't be detected and automatically configured under a modern operating system. The instructions on the SuSE support site said to add lines to lilo.conf and reboot. While this is a perfectly acceptable way to get hardware working for a geek familiar with *NIX, I believe that a home user shouldn't have to do more than plug it in. It's an IDE device, it's not that complicated!
The ugly: Once the hardware was finally working (as a pseudo-scsi drive), the next hurdle was to find decent graphical tools to burn and copy CDs. I finally settled on CDBakeOven, an above average KDE application. It burned CDs from data on the hard drive, but for some reason cdrecord (the command line backend) refused to allow me to copy a cd directly. Yes, it was installed SUID root. CD copying is such a basic function nowadays, why is it so hard to do under GNU/Linux?
Software distribution
I'll put this simply. I'm a home user, not a programmer. Why on earth should I have to compile the software I want to use? I know that having the source available is a good thing, but I'll say it again: I'm no programmer. I just want to install software and run it.
This leads to another point. Although having package databases (such as the rpm and deb systems use) is great, there should definitely be seperation between system packages and additionally installed software. There needs to be a standard installer and database for user-installed applications such as word processors, email clients and games, and it should be seperate from the rpm or deb databases used for system software such as lilo, init and cron. This will make it much easier for home users to know what applications they have installed on their PC, and to easily uninstall them if necessary, without knowing some arcane commands and weird package names.
Support
There is a huge wealth of knowledge among the thousands (millions?) of people that run GNU/Linux around the world. If you have a problem, odds are that someone out there can help you, often for free. This is one of the linux platform's greatest strengths. However, Linux users are also its greatest weakness. This may not apply to most of the community, but there is a very vocal minority that gives Linux a bad name. To every Linux user that has ever helped a newbie, I thank you. I have been helped by many a guru, often when I've been asking the simplest of questions. It's the remainder that are a problem.
I once heard a song by Three Dead Trolls in a Baggie called Every OS Sucks, where Linux users were described as 'elitist nerdy shmucks'. Sadly this is true for much of the 'community'. Too many consider themselves better than the rest of the world because they run Linux. Can you believe that? It's just a computer operating system, but somehow they think that it makes them better than those people who run systems such as Microsoft Windows! Elitism drives people away, as does saying âoeRTFMâ or belittling people who choose a different distro from yourself.
'Nuff said about that.
So what now?
Well, I decided to go back to a Microsoft platform. Initially being paranoid after reading things about DRM and spyware, I bit the bullet and installed Microsoft Windows XP. Like every OS, it has good and bad points; most of which you can learn about from online reviewers. I'll just point out several things that make me want to keep using it instead of GNU/Linux.
Fast graphical subsystem: Windows has lighting quick graphics, both 2d and 3d. There's no denying it. When I move a window, it refreshes so fast that I don't miss X11 at all. While not quite as nice as some other operating systems, font support is outstanding compared to XFree86.
Drivers: Point and click to install (as a superuser, of course). Windows warns you if the driver isn't likely to work properly, and can roll back to working drivers if you deliberately choose to install one that hoses your system.
Hardware setup: My CDRW worked right away, without a hitch. I am able to drag and drop files from the Explorer file manager to the CDRW icon and they get added to the list of things to burn. A quick install of Nero Burning Rom, and I was able to make a backup copy of my game CDs. (I don't like taking originals to LANs where they can get destroyed or stolen).
Software distribution: All windows software comes in binaries, either with an installer or in a zip file. I hope to never compile an application ever again. Software designed for a different version of windows is 99% guaranteed to run, but if not, there is always 'compatibility mode'. One thing to note, however: Applications designed for single user versions of windows usually only run properly as a superuser, and this includes 3d games. I expect this to be rectified as the rest of the Windows world catches up to a multi-user environment.
I can't comment on the Windows using community yet. I've not yet had a problem that a simple point and click couldn't fix. However, I will say that my original concern with Windows '95 has been addressed in Windows XP. The stability is finally there.
Final Notes
In conclusion, I'd just like to make it known that I haven't completely abandoned the Linux community. My home server still runs Mandrake, and IPCop on my gateway/firewall. There is no way I'd ever put any form of Windows on my server, nor would I ever connect a Windows PC directly to the internet without a *NIX gateway in between. Microsoft has a history of poor security, so I protect myself the only way I know how; using Linux. I will continue to advocate the use of GNU/Linux in the server arena. This is where its strength lies at the moment.
Because of their history of spreading virii, I don't use the applications that Microsoft has provided with Windows XP. My wife and I use Mozilla for web browsing and email, OpenOffice.org for word processing, and Psi (Jabber client) for instant messaging. All of these are true multi-user win32 programs, and are perfectly interoperable with their Linux counterparts.
I expect that the Linux community will have something to say about this article; I welcome comments and constructive criticism. Flames will be automatically sent to the Windows equivalent of
By Tony âoekNIGitsâ Collins
- posted by poopbot: providing truth in a deceitful world
2TloJkTYEM Post #348
Q
W
E
R
T
Y
U
I
O
P
A
S
D
F
G
H
J
K
L
Z
X
C
V
B
N
M
Q
W
E
R
T
Y
U
I
O
P
A
S
D
F
G
H
J
K
L
Z
X
C
V
B
N
M
F
U
C
K
!
You heard me. I was going through security at Fort Lauderdale airport, and this guy was really scrutinizing me. Fine, I had no problem. Until he reached down my pants. He said it was because of my belt buckle. The problem is, the security screeners barely speak English. For people who do this all day, they are pretty bad at giving clear instructions. He mumbled something, and I thought he was telling me to sit down. So I sat down. Then he said "no no no, stand up." He went over me with the wand, and my belt buckle set it off. Then he asked me to undo my belt, and he stuck his hand down my pants! So I said, "What the hell are you doing?" He didn't have an answer, he mumbled something. Fucking people responsible for our security should not be illiterate perverts. I think it's about time to dump the bar. I am sick and tired (and tired always follows sick). But that's a small thing in the end, I guess. Right now, I'm waiting to find out if I have to do the jury duty thing. I won't find out until 11:00 or so. Until then I can "entertain" you guys with my weekend. Bear with it. There's light at the end of the diary... So, Friday night felt like a complete clusterfuck. I was down a man (a family reunion in Abilene, poor guy) and training 2 new guys. It just felt like everything that could go wrong went wrong, but that wasn't the real topper to my weekend. Friday night, one of my Underlings (specifically Jigsaw, frequently referred to as Underling1) told one of the managers and told me that he would be calling in on Saturday and wouldn't be coming in. You see, Jigsaw asked me a couple of weeks ago to get Saturday off for a family thing. I couldn't give it to him because the other guy who was off asked first, gave me more then a month's warning and I could only spare one person. So Jigsaw just decided to "tell" me that he wasn't coming in. Specifically, he said that no job was worth disappointing his family. Noble sentiment that was going to piss me off. So Saturday night, I had one experienced person, two newbies and myself working. It was a little packed but manageable for the first couple of hours until "They" arrived. And by "They," I mean Vinny and Dimebag from Pantera. That's when the pooch got screwed. People started calling all their friends to come to the bar because Vinny and Dimebag were there. People started drinking because they were hanging and being bad asses with Pantera. Truth be told, I couldn't give a fuck about them, but when celebs show up, it's my job to take care of them. Being a man short made that job difficult, but doable. Being another man short made it next to impossible. But the real topper was that there was a fight in another part of the club and I didn't even find out about it until it was long over and done with. Turns out that one of my Underlings went to toss a guy, and the guy turned on him and none of us (Security) was there to back him up because we were spread so thin. So a couple of the bartenders came from behind, got the guy out, told him to go away, got swung at and chased him into the street to get him gone. I played no part in this at all because while all this was happening, I was at the door carding people. If Underling1 had been there, I wouldn't have been at the door. I would have been able to go where the trouble was. I was not happy at the end of Saturday night. So I took Underling1 off the schedule for a couple of days. Did I act hastily? Maybe I did, but I'm not going back on this decision. I could say some kind of thing about people being the fingers of a hand or the cogs of a machine, but all that's crap. What it comes down to is one of my people let me down. It wasn't even because he was sick or something. I know it was a family thing, but I have bent over backwards to accommodate him and his family. I figure just because I don't have a good relationship with my family doesn't mean I should begrudge him his. But I've done it far too many times. And he's just going to inform me that he's not coming in? Well, I can inform him that he's not coming in also. Truth be told, I should have fired his ass, but unfortunately, I need him because we're already short of people and he's going to fuck up and get fired eventually because he's done it before. The thing that really pisses me off is that his absence kept me from doing my job as well as I could. And that's why I should probably drop the bar. I can't do the job as well as I would like because I'm only working there two days a week and it's not worth the money or the risk to my person. That and I have a lot of trouble caring about some facets of the job, too. But I seem to have an overdeveloped sense of responsibility. I don't want to leave them in the lurch. I've already had people tell me that when I leave, they think things will go back to shit. I don't have the hubris to think that the entire place will fall apart without me, but I know that I do a lot there and they'll never get another person like me anytime soon even if the double what the position pays. I should be replaced and soon. I did get one uplifting thing this weekend. A friend of mine and his wife finally legally changed her son's last name to his (and her married) name. They finally gave him a middle name, too. And the named him after me. Aeren Marcellus Throgmorton. I was and am completely touched. That just kinda makes all the other stuff just not matter doesn't it? This is without a doubt the biggest honor of my life. Heck, this even beats when Bob asked me to be his Best Man. I rule. I really do. In this essay we shall explore an alternative to the use of vector algebra for representation of physical objects in the sciences. It is not the purpose of this essay to make converts of the readers. It is enough to demonstrate how geometric algebras can be used to do many of the same tasks as the vector algebras along with a few additional ones usually reserved for other mathematical tools. In a previous work, the topic of how we choose to represent physical objects in our theories was discussed. Three postulates were laid out defining the concepts of completeness, rendition, and identity. This work will focus mostly upon the second term by offering geometric algebra as an alternative to vector algebra and matrix algebra. A Brief History The concept of a geometric algebra can be traced back to 1797 to a Norwegian surveyor named C. Wessel who interpreted the unit imaginary number as a directed line segment perpendicular to the unit real number. Making the complex numbers geometric has proven to be quite fruitful in a number of fields In 1843, W.R. Hamilton created the algebra of quaternions as an extension of complex numbers with the intent to represent three-dimensional physical objects. Around the same time, H. Grassman invented exterior forms. His algebra would eventually be developed into differential geometry. Both of these tools are powerful, but Grassman's contribution is better known today. In 1878, W.K. Clifford generalized and reinterpreted the algebras of Hamilton and Grassman. Some will refer to one of the algebras as biquaternions, but in their general form they are named geometric algebras. Clifford's algebras restricted themselves to simpler definitions for multiplication and addition similar to Hamilton's approach but including Grassman's exterior product within the general product. These operations were defined that way in order to obtain geometric meaning. Clifford died of tuberculosis a year later. Clifford's results have been rediscovered a few times by more recent researchers. Sauter, Sommerfeld, and Eddington all noticed that Dirac's gamma matrices were generators of a four dimensional geometric algebra with a space-time metric. These are the gamma matrices from Dirac's first order relativistic theory for electrons. Several other authors have noted that classical electrodynamics can be easily written in the same geometric algebra generated by a suitably defined set of gamma matrices. In the early 1880's, J.W. Gibbs wrote about his vector algebra. Gibbs' algebra and his notational approach were a limited form of Grassman's ideas. The system was adopted by those developing the theory of electrodynamics at the time and won the battle for the approach taught to students. The vector algebra taught to students today largely derives from Gibbs' system, though the notation has been changed a little. Between 1891 and 1894, in the journal Nature, long letters defending various notions can be found between the proponents of both sides of the representational conflict. On one side were Gibbs, and O. Heaviside. P.G. Tait supported the other side. By this time, the main creators of the quaternionic approaches were long dead. The creators and proponents of the vector algebra were alive and well. The actual argument in the journal was a little lop-sided but it went on outside the print media for some time. Anyone interested in what a 19th century flame war looks like is encouraged to do a little digging in the library for volumes 43 through 46. Gibbs first letter was published April 2, 1891 and Tait's first response appeared four weeks later. Definitions To start, we offer one formal and one informal definition of a geometric algebra. The formal one has a few parts and is meant for those readers with the mathematical background to understand them. In the truest mathematical sense, this definition could be made more formal. This version will suffice for this work. The informal definition, however, is the one most likely to help build an intuitive understanding of how to use these algebras as tools. Formal Definition of a Geometric Algebra 1. Binary Operation A binary operation on a set is a rule that assigns to an ordered pair of elements of the set some other element of the set. 2. Group A group is a set together with a binary operation on that set such that the following holds true. * The binary operation is associative. (a.b).c = a.(b.c) * There is an identity element in the set relative to the operation. * For every element in the set, there is an inverse element such that the operation on both of them always produces the identity element. (An abelian group is one where the binary operation happens to be commutative. a.b = b.a) 3. Ring A ring is a set together with two binary operations (called + and *) on that set such that the following holds true. * The set and the + operation form an abelian group. * The other operation (*) is associative. * The * operation is distributive over + from the right or the left. a*(b+c)=a*b+a*c and (b+c)*a=b*a+c*a 4. Field A field is a commutative ring under * with a unity element under * where all elements except the + identity have * inverses in the ring. 5. Vector Space A vector space consists of an abelian group under an addition-like operation, a field, and an operation (x) between elements of the field and vector space where the following holds true. * (x) produces and element of the group. * (x) is associative * (x) is distributive in the field AND in the group * There is an identity element within the field for (x) 6. Algebra An algebra consists of a vector space together with a binary operation (.) on the elements of the group such that the following holds true. * (.) is associative with elements of the group * (.) is commutative with the scalar multiplication of the vector space. * (.) is distributive with respect to the + operation in the group on the right and left. (Generators of an algebra are a set of elements such that all possible products produce a basis for the associated vector space. Linear combinations of the basis elements cover all possible elements of the algebra.) 7. Geometric Algebra A geometric algebra is a type of algebra with a set of generators where the following holds true. * products of two different generators under (.) anticommute * products of two identical generators under (.) are defined (The generators of a geometric algebra are the elements to which geometric meaning is attached.) That is a formal enough definition along with the terms that provide the underpinning that makes the definition work. There are two multiplication-like operations, a single addition-like one, a set of elements that starts this all off, and some rules they must all obey regarding commutativity, associativity, distributivity, and a variety of types of operational identity elements and inverse elements. This cloud of formalism is important in the mathematical sense. For our purposes, however, it is more instructive to approach the tool informally and discover these definitions through usage. Remember the existence of these formal definitions, though. Some of the seemingly magical qualities of geometric algebras discussed later can be shown to be obvious by the fact that they are designed into the tools from the start. Informal Definition of a Geometric Algebra Start with a set of objects we choose to use to represent directed lines. We will call them vectors later, but for now they are generators. With these generators, we shall construct everything in our geometric algebra. Note the following. 1. The number of generators in our set defines the meaning of 'dimension.' 2. The field for our algebra will be assumed to be the Real numbers unless otherwise stated. 3. Experience the reader already has regarding the operations + and x from vector spaces will continue to apply here. The concept of 'linear combination' still works here too. 4. We shall define (.) by providing the multiplication table involving all generators and their products. 5. All elements found on the (.) multiplication table form a set that is sufficient to span the vector space found within the algebra. 6. With (.) we introduce the term 'algebraic combination' as an extension of linear combination and allow both (.) and (+) operations. A two dimensional example. Let's call our generators X and Y. We need to define a multiplication table for the (.) operation to be defined. Here it is written out. X . X = 1, X . Y = XY, Y . X = -XY, and Y . Y = 1 With this list, we have a two-dimensional algebra where 1, X, Y, and XY span the related vector space. Elements of the geometric algebra can be represented as linear combinations of the set that spans or as algebraic combinations of the generators. Here is an example. (Remember that x is the scalar multiplication with the field.) M = (A x 1) + (B x X) + (C x Y) + (D x XY) 1 is interpreted as a point. The field element 'A' changes its magnitude. X and Y are interpreted as directed line segments. (The vectors) Remember that a directed line is a one-dimensional object with a sense of forwards or backwards. XY is interpreted as a directed plane segment. (A bivector) A directed plane is a two-dimensional area with a sense of rotation clockwise or counterclockwise. A three dimension example. Let's call our generators X, Y and Z. The multiplication table for the operation is as follows. (Assume a.b = -b.a where a, b are any generators.) 1.X=X, 1.Y=Y, and 1.Z=Z X.X=1, Y.Y=1, and Z.Z=1 X.Y=XY, X.Z=XZ, and Y.Z=YZ XY.X=-Y, XY.Y=X, and XY.Z=XYZ XZ.X=-Z, XZ.Y=-XYZ, and XZ.Z=X YZ.X=XYZ, YZ.Y=-Y, and YZ.Z=Y XYZ.X=YZ, XYZ.Y=-XZ, and XYZ.Z=XY XY.XY=-1, XZ.XZ=-1, and YZ.YZ=-1 XY.XZ=-YZ, XY.YZ=XZ, and XZ.YZ=-XY Elements in this algebra can be written as follows. M = (A x 1) + (B x X) + (C x Y) + (D x Z) +(E x XY)+ (F x XZ)+ (G x YZ)+ (H x XYZ) XYZ is interpreted as a directed volume segment. (A trivector) A directed volume is a three-dimensional object with a sense of rotation (left-handed or right-handed) around a parallelepiped. (The limitations of HTML or the author's knowledge of HTML in K5 articles makes this table somewhat hard to read. Imagine a square table with the eight elements running across the top and down the left. Fill in the products like one was taught to do with multiplication tables involving real numbers to get the table described above.) Other examples are possible and worth considering. The reader is encouraged to think about how the multiplication tables would change in both examples if X.X = -1 instead of +1. Such a change makes one of the generators begin to behave like a time-like vector instead of a space-like vector. A later example will show a four-dimensional algebra where one of the generators has a negative square. Such an algebra begins to behave like a Minkowski space, though it is geometrically richer. If both X and Y, in our two-dimensional case, were changed to produce negative squares in the two dimensional example, we would reproduce the algebra of quaternions. Why learn another paradigm? A reasonable concern for anyone currently using vector algebras and not familiar with geometric algebras is whether this new approach to representing objects is worth learning. The answer to this concern comes in two forms. The first is that an alternate approach to the representation of objects uncovers some of the techniques we use in rendering those representations. The second comes from the fact that some of the elements of our geometric algebras do not have analogs in the vector algebras commonly used to represent physical objects. These unusual elements may be worth exploring in case they help uncover new mathematical structures that imitate unexplained experimental evidence. Geometric algebras permit the sum of objects of different geometric rank. What is a vector + plane? Readers familiar with vector algebras know that such an object simply can't be constructed without violation of the transformation rules that maintain the concept of identity. Geometric algebras lead to no such violations, so there is a difference. Representation Techniques The interpretations for object within a geometric algebra depend on their rank and on their position relative to one of the operations. This is very similar to how we do things in other mathematical tools. A simple example will be shown for real numbers and then expanded to cover a three dimensional, Euclidean geometric algebra. Consider the equation (2 times 3 is 6.) The binary operation is the familiar one of multiplication of real numbers. This same equation could be written as times(2, 3) is 6. Any programmer would recognize the operator and the operands as distinct. However, because the operands of a binary operation are ordered, we could also write the equation with a unary operation as double(3) is 6. If the first operand of the binary operation were held constant, the unary version would be equivalent and more efficient to boot. This example shows that the operand can also be considered as part of the operation. This ambiguity is the one that leads to two possible interpretations for the same thing. One version treats the object as an input while the second treats it as an operator. In a three dimensional Euclidean algebra, we have three generators named X, Y, and Z. From these we can create 5 other objects using the multiplication operation alone. The objects are {1, X, Y, Z, XY, XZ, YZ, and XYZ}. Their interpretations as operands are relatively simple. Each is a directed element of rank n where n is equal to the number of generators it takes to represent the object. 1. The object '1' is of rank zero and represents points with magnitude. It is not the real number we usually label with a '1' symbol. 2. X, Y, and Z are directed line segments similar to basis vectors. 3. XY, XZ, and YZ are directed plane segments 4. XYZ is a directed volume segment. Note that no matrices are necessary for these interpretations. Each of these eight objects and their related interpretations is as 'real' as any interpretation of the unit imaginary number. The interpretation of these eight objects as operands takes a bit more work. 1. The object '1' is a scaling operation. By itself, it scales other objects by unity. When it is scaled by a real number from the Field, it transfers that scaling to the operand. 2. The generators are reflection operators if the operation is done in a bilinear fashion. Consider a linear combination of the generators. V = a x X + b x Y + c x Z. If you multiply on both sides by X, the result is XVX = a x X - b x Y - c x Z. Notice that the parts of V that were perpendicular to X changed sign after the operation was done from both sides. 3. If the generators are reflection operators, then the bivectors must perform two reflections. It is possible to write two reflections about different lines as a single rotation, so the bivectors when operated in a bilinear fashion are rotation operators. 4. The last one, XYZ, is a parity operator. All three generators get reflected, but the affect cancels out on the bivectors. With these two types of interpretation, we can address physical situations. Consider a room full of warm air. We could represent the temperature at every location in the room with a function of scalars (1's) that are scaled to match our readings. We could represent the movement of air molecules with a function of vectors (linear combinations of generators). The size of each directed line is linked to the momentum of the molecule. We could represent the spinning of each molecule with a function of bivectors. The magnitude of each directed plane is linked to the angular momentum of the molecule. In a vector algebra, we would keep these functions separate in order to avoid violations of the transformation rules that define identity. In a geometric algebra, this distinction is not needed. No matrix representation of the generators and the other objects is needed for the geometric algebra version as long as we keep the multiplication table handy. The matrix form can also be avoided for the vector algebra version, though it is seldom done. Our freedom to add objects of different ranks in a geometric algebra leads to some unusual constructs one does not normally encounter if one sticks with vector algebras. Consider the following objects. M = 0.5 x ( 1 + X ) and N = Y ( 1 + X ) The object M has the curious property that when squared you get it back again. The object N has the curious property that when squared you get the additive identity named 'zero.' M is an example of a set of elements of the algebra referred to as idempotents while elements that behave like N are referred to as nilpotents. These constructs are not possible in vector algebras, so if there is anything that could support or demolish the usefulness of geometric algebras as a representational tool in physics theories, it will probably lie with these constructs. Examples An example or three will demonstrate just how similar geometric algebra can be to the current vector algebra. The rendering of objects in both systems is very close and easily confused. The difference all rests in how the reference frame is written when rendering real objects. Example 1 Imagine our room full of warm air again. Each molecule in the room is moving around at some velocity and, therefore, with some momentum. Imagine tagging each molecule with a directed line segment representing that momentum. With a little sleight of hand we could pretend your tags actually referred to a small volume of some continuous fluid and avoid the discrete nature of the air. This step isn't strictly necessary, but it does make writing a function for the momentum of the fluid easier. That function would appear as follows. Momentum density P (position) = sum [coefficient of P x a basis element] where the basis elements are the eight objects that can be generated from the basis vectors. Obviously, most of the coefficients will be chosen to be zero in order to ensure that P is a vector density. That works just fine. In a vector algebra we would write the same equation as follows. Momentum density P (position) = {Px, Py, Pz} where the coefficients are part of a 3-space vector. The reference frame to which those coefficients apply is implied in the equations and usually gets described somewhere nearby in the text explanation. Some people will start with an equation that looks like the first one but limit it to a sum over only the vector coefficients. Then, when a calculation must be performed, they switch to the matrix notation for vectors and proceed. These people are living halfway between the two systems and probably find some intuitive advantage to the geometric approach whether they know they do or not. Example 2 For another example, imagine a planet orbiting the sun. That planet is at a distance R and moves with a momentum P. We shall write the angular momentum of the planet using both approaches. In the vector algebra, we would use a cross product defined by the right hand rule to get the following. L = R X P (R cross P) L is a vector defined as perpendicular to R and P at the same time. Since there are two possible directions for this 'vector', the right hand rule is used to choose the right-handed one. The magnitude of L is the product of the magnitudes of R and P times the sine of the angle between them. It turns out that the cross product defines a different kind of vector from R and P. It is important for users of this system to distinguish polar vectors from axial vectors, as they are not really the same thing even if they appear to be. In the geometric algebra, we would use the general product defined in the algebra to get the following. L = R ^ P = 0.5(R.P - P.R) (^ is the antisymmetric part of (.) in this case) The geometric nature of L is dependent on the (.) product and not defined to be a vector. As it turns out, L is a bivector. In a three dimensional, Euclidean space, L has the same number of coefficients whether it is described as a bivector or as a vector, but its nature under parity transformations is quite different. A little more exploration would show that all the axial vectors from the other system are bivectors in a geometric algebra. Keeping the polar vectors separate from the axial vectors is done automatically since they are of different ranks. Example 3 The last example involves an operator that rotates other objects. This example will be restricted to two dimensions in order to make it easier to understand. There is only one plane in two dimensions, so there is only one kind of rotation operator. Imagine a two dimensional vector described as V = A x X + B x Y. Suppose we want to rotate it by 45 degrees swinging the X direction up toward the Y direction a bit. The left sided operator that does this is as follows. R = cos(45) x 1 + sin(45) x XY Multiply them as follows. Vprime= R.V = * = (cos(45)1+sin(45)XY)(AX+BY) * = cos(45)AX + sin(45)AXY.X + cos(45)BY + sin(45)BXY.Y * = cos(45)AX - sin(45)AY + cos(45)BY + sin(45)BX * = [cos(45)A+sin(45)B]X +[cos(45)B-sin(45)A]Y Thus far, we have shown enough material to demonstrate how to use the tool for classical theories of physics. With a bit of practice, the reader could use geometric algebras to see why special relativity is built into the fabric of their solution space if they pick the right algebra and the right metric. For the physicists, though, this is not enough. A better mathematical tool to do what we already know how to do is only of pedagogical interest. Whether geometric algebras provide engineers and software developers more efficient approaches to writing solutions remains to be seen. For the physicists and physics students, then, we shall provide one more example that needs more work on a theoretical and experimental level. To do this, we must provide one more definition. 8. Left Ideal A left ideal is a set of elements from a ring that behave as follows. If M is an arbitrary element of the ring and L is an element of the left ideal, then M.L is an element of the left ideal. Our geometric algebras are rings, so a left ideal in one of them has the property that any element multiplied from the left with an element of the ideal is still in the ideal. This has the curious property of ensuring that operators that can be written in the same algebra as the operands will not take you outside the left ideal if you start there. Why this is interesting to the physicists is best shown by an example in a four dimensional algebra with a space-time metric. Example 4 Consider an algebra whose generators are CT, X, Y, and Z. Let the first generator have a negative square while the others have positive squares. Also let the algebra be defined over the complex numbers for the field. From this algebra it is possible to write an idempotent that looks as follows. I = 0.25 (1 + iCT)(1 + CTXY) or = 0.25 (1 + ip/m)(1 + E5.S) where p is a time-like four-vector, S is a space-like four-vector and E5 is the quadvector. It turns out that if you try left multiplying any element of the algebra against I you will not cover all the possible elements in the algebra. Only a subset can be reached through left multiplication, so I can be said to generate a left ideal. In a four dimensional algebra, we start with four generators and create sixteen basis elements through multiplication. This sixteen-element set is the basis of the more familiar vector space associated with the algebra. If we use our idempotent described above, we can create a smaller vector space spanned by four elements that can be written as follows. Basis elements of the left ideal are 0.25(1 +/- ip/m)(1 +/- E5.S) Anyone with any exposure to early relativistic quantum theory will recognize this ideal as having the same 'states' as a four-component bispinor. Since these elements span a left ideal, left-sided operations on them from other elements of the algebra ensure the resulting element is still in the ideal. This is very much like saying that an electron is an electron no matter what state you happen to find it in. Therefore, the left ideals ARE renderings of the particles that an algebra can describe. No appended spinor space is required to get spinor-like behavior from these algebras. The behavior is already there. The really cute thing is that spinor-like behavior can also be found in the classical theories written with these algebras. There is no way to get it out, really. Future Work The bulk of the work that is occurring today using geometric algebras is being done at the post-graduate level by academic researchers. Their results get discussed at conferences and published in peer-reviewed journals. As a consequence, it is not that easy for someone outside the immediate field to break in as a beginner and learn to be productive. This is mostly due to the fact that the physics problems being research are complex, post-graduate problems. The usage of the geometric algebras is not all that complex. There are two open source projects managed by the author that make use of geometric algebras. The first supports the creation of a java library to be used much like a library that supports matrix and tensor objects and their operations. The second project is a solar sail simulation application that makes use of the library from the first project. Anyone curious about the topic of geometric algebra is invited to check out these projects and see geometric algebras in action. Clados Project = http://clados.sourceforge.net SailAway Project = http://sailaway.sourceforge.net References Representation of Objects: http://www.kuro5hin.org/story/2002/5/7/1345/25152 Clifford Algebra Society: http://www.clifford.org D. Hestenes Book: New Foundations for Classical Mechanics. ISBN 90-277-2090-8 D. Hestenes website at ASU: http://modelingnts.la.asu.edu/GC_R&D.html Cambridge Group: http://www.mrao.cam.ac.uk/~clifford/