Mathematica 6 is a computational engine par excellence. It can handle just about any kind of complex math you can throw at it. It's for the big dogs (mathwise). Colleges and universities use it to further science and mathematics as we know it -- and with a pedigree like that, it's got to be good. And it is.

What Can It Do?
When you start Mathematica 6, the first thing you'll see is a blank window that Wolfram calls a notebook. You can add all kinds of things to the notebook to use in your computations. Mathematica 6 uses dynamic interactivity to create equations and other devices to solve for what you need. You can apply buttons or sliders for values (see figure below).

Mathematica 6 lets you add sliders (that's neither the TV show nor the sandwich) to your notebook. You can graphically vary your parameters to affect your computations directly. (Click image for larger version)

That way you can vary the values in your equations. You can even tie the values to a dynamic table. You can import values from other programs to populate your tables in a wide range of formats. Interestingly, you can add graphic elements to indicate variables. (You can even draw in a squiggle to use as a variable.) When you're done, you can export your data to literally hundreds of file formats -- 3DS, AVI, BMP, MIDI, etc. -- in all sorts of applications. One thing I thought was cool was that you can import sounds and then chart and manipulate them. That ability is for when you want to interact with real-world data.

Mathematica 6 allows you to pull in GIS data and use it for graphics. And you can manipulate it, too. Why would you need to do that? Well, what if you are building a dam? You might want to see how much change would be required to your building site to affect a given amount of water. This calculation would be fairly easy with Mathematica 6. You have support for 3DConnexion devices (which I think are great!). They allow users to rotate their graphics and scroll through their notebooks like champs.

Another great function is 3D printing. Not only can you create some remarkable graphics with Mathematica 6 (see figure below), but you can export them to a 3D printer to create parts that you can hold in your hand. Nothing says visualization like a hard-copy part! You can actually create mathematical models that would be difficult, if not impossible, to build in any other way.

You can set up some pretty complex mathematical objects. By changing the variables and parameters the shape can change drastically.

Users can download a free Mathematica player so they can distribute any notebook created in Mathematica 6 to whomever they want. With the player, you can view the notebooks, play animations, listen to sounds, and copy information to all sorts of other software applications. Depending on how you export the notebook file, you can even use the sliders and buttons to manipulate the data included. That way you can show your boss all the things you went through to arrive at the solutions you did. They can go through the what-ifs and see what you saw. Of course, to make permanent changes they would need the full installation of Mathematica 6.

Mathematica 6 has plenty of add-ons: webMathematica, which puts your information on the Internet or a network for interact with others; gridMathematica, which takes advantage of multiprocessors and networked resources for extreme high-end computations; Mathematica Personal Grid edition, which uses modern multicore parallel processing; and more. Wolfram offers online demonstrations so you can get up and running as soon as possible.

What's Wrong With It?
One area of concern with Mathematica 6 that I imagine might be problematic (at least for folks like me) is the input paradigm. In this age of graphical interfaces and click-and-drag, Wolfram has chosen to use text entry for input. Not that there's anything wrong with that; I'm sure there are a lot of people out there who prefer it to a more modern, graphical way of doing things. Does it sound bad to say it that way? Well then, let's not beat around the bush. Mathematica 6 may be ultrapowerful and bristling with functionality, but in my opinion using a text-entry interface places it right back in the early 1980s (you know, last century, power suits, Devo?). You have to type in commands and remember to keep your syntax right. Take a look at the sort of thing you will find yourself typing (see figure below).

I don't know about you, but this isn't even Greek to me. The line of text across the top tells Mathematica 6 what you want the graph to do.

I felt like I needed to be a programmer! Maybe I'm being a little harsh, but after looking at Mathematica 6's competition and the way the others are heavily graphically oriented, I think these are valid comments. Not to worry, though, if you're typing along and you just can't remember that text-string command, Mathematica 6 will help you out. If you can remember just a part of a command, you can find it (see figure below). Mathematica 6 will offer you a menu to pick your commands from based on the fragment you type in. All you have to do is know what it is that you want to pick. You also can enter equations graphically through the palettes.

Figure 4. Undoubtedly in an effort to help the memory-challenged among us (me), Mathematica 6 will offer possible commands when you can't remember what you want. (Click image for larger version)

Mathematica 6 is professional software and is priced accordingly. The cost for a professional license is $2,495 and includes a one-year subscription for technical support, upgrades, and so forth. Volume discounts, site licenses, and academic discounts are also available.

Weighing the Pros and Cons
Mathematica 6 is one of the most powerful computational software packages I've ever looked at. It stands ready to solve whatever math problem you have. With its text-entry interface, I don't suppose it's really going to be super-easy to learn to use, but if you are planning to plumb the depths of its functionality, my guess is that you're not really looking for easy anyway. For more information about Mathematica 6 or other products, visit the Wolfram Web site.