“Bursting with technology” is how Stephen Wolfram described Wolfram Research during his opening keynote at the Wolfram Technology Conference in October.

The recent release of Mathematica 8 marks the beginning of a whole new way to compute and program thanks to its free-form linguistic input. Additionally, the ever-growing Wolfram|Alpha computational knowledge engine has doubled in content since its launch in May 2009 and continues to become more ubiquitous in the world.

Stephen says you will soon “see a lot of different directions emerge” based on the technology and technology platforms that Mathematica and Wolfram|Alpha provide.
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Here is a shot I took of M27, the famous Dumbbell Nebula, with my home-brew 90mm astrograph and inexpensive equatorial mount.

Actually, it isn’t a single shot, but a combination of about 30 fairly short exposures, added together. Adding together short subframes rather than taking a single longer exposure makes it possible to create astrophotos without additional equipment for “guiding” the telescope. Guiding means applying small corrections, either manually or automatically, during the exposure to compensate for imperfections in either the mount’s alignment away from the polar axis or the mount’s drive mechanism. Combining the subframes has the additional benefit of reducing noise and increasing the signal to produce a result similar to a much longer exposure.

Before we go further, it’s fun to look up information about M27 using the new Wolfram|Alpha features built in to Mathematica 8.
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Regular readers of the Wolfram Blog will have seen that the item that I wrote on solving mazes using morphological image processing was thoroughly beaten by the much smarter and better read, Daniel Lichtblau from our Scientific Information Group in his post “Navigating the Blenheim Maze”.

Always up for a challenge (and feeling a little guilty about the rather hacky and lazy way I tried to deal with loops and multiple paths the first time), I am back for another go.

My first approach with any new problem is to think about the nearest available Mathematica command. In the new Mathematica 8 features is a graph theory command FindShortestPath. That certainly sounds promising.

Mixing image processing and graph theory may sound complicated, but one of the central strengths of Mathematica’s integrated all-in-one design is that different functionality works together, and in this case it proves to be quite easy.
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Your assignment:

Write a simulation of spherical particles coalescing under gravitational attraction. Limit the approach distance by a secondary repulsive force that acts over short distances. Produce an animation of the dynamic system starting with 15 particles in randomized positions.

Formulate your solution in 140 characters or less.

Sound challenging? A 138-character solution was Stephan Leibbrandt’s winning entry in the Mathematica One-Liner Competition that was a part of this year’s Wolfram Technology Conference.

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Editorial note: A future post will explore some of the contributions to the visual arts and media facilitated by Mathematica.

The year 1982 saw a lot of important movies: Blade Runner, E.T.: The Extra-Terrestrial, Poltergeist, Star Trek II: The Wrath of Khan, The Thing, Mad Max 2: The Road Warrior, Pink Floyd The Wall, First Blood, Conan the Barbarian, Fast Times at Ridgemont High, The Dark Crystal, and TRON.

I used Mathematica functionality to turn the TRON logo into something you can manipulate. You can download my notebook to play with the logo. (Mathematica Home Edition could be used to do this as well.)

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I just published a Mathematica package that provides an alternative, richer implementation of units and dimensional analysis than the built-in units package. You can get it here. Aside from being a really nice extension to Mathematica, it is also an interesting case study in adding a custom data “type” to Mathematica and extending the knowledge of the built-in functions to handle the new “type”.

First I have to explain the point by answering the question, “What’s wrong with the built-in units package?” Well, there is nothing actually wrong with it, it just doesn’t apply Mathematica’s automation principles. It can convert between several hundred units and warn if a requested conversion is dimensionally inconsistent. But give it an input like…

and it does nothing with it until you specify that you want the result in a specific unit. The core reason is that it doesn’t teach the system, as a whole, anything about units, or even that the symbol “Meter” is any different than the symbol “x”. All of the knowledge about units and Meter in particular is contained in the Convert command.
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I read Jon McLoone’s recent “aMazeing Image Processing in Mathematica” post with some interest.

It showed how to import an image of a maze, and then use image processing functions in Mathematica (some new to Version 8) to draw paths through the maze. What fun! I then observed, to my dismay, that there was no way to determine a “good” path. Frankly, I was disappointed.

I decided that there must be ways to do this in Mathematica. One approach would involve forming a graph. We would have vertices at points where the maze path forks, and we would make weighted edges from approximated distances between these vertices. New functionality in Mathematica supports these graph methods. Unfortunately I am not yet familiar with it.
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