Monthly Archive: March 2019

The year was 1995.  A short-lived and largely unnoticed television show premiered on Fox from March to May of that year.  The show centered around a young woman named Sydney Bloom (played by Lori Singer, who played the role of Cassandra in Warlock), whose father bequeathed her a strange ability.  She was able to enter VR.5, the fifth level of virtual reality, which consisted of her entering with total perception into a cyberspace world that sported realism that matched our own day-to-day reality.

The premise and promise of the show was that a fully immersive digital world was just around the corner.  All that was needed was the familiar trappings of eye-gear, power-glove, computer, and cheesy special effects.  Those modest ingredients allowed her to totally connect with a different reality or, at least, another facet of our reality.

Just a scant three years earlier, the entertainment industry subjected the film-going public to the cringe-worthy spectacle that was the horror movie The Lawnmower Man.  Loosely based on a Stephen King movie, the story follows the tragedy that ensues when a scientist (played by Pierce Brosnan) attempts to boost the brain power of a developmentally challenged man (played by Jeff Fahey) through a combined use of drugs and virtual reality.

Here we see a fairly early representation of tactile suits that are designed to simulate the sense of touch, as well as the concepts of an avatar to represent one’s persona in the virtual world and the reoccurring theme of megalomaniacal behavior brought on by unfettered cyber-license.

Without a doubt, the most esteemed member of all virtual reality stories must be The Matrix.

Premiering in 1999 under the shadow of The Phantom Menace, the first Star Wars movie in over 15 years, The Matrix was largely overlooked in the months leading up to its release, but the public reception of its storyline and the philosophical questions that it raised made it a classic of American film.  It set the bar for what virtual reality would mean in the science fiction genre, and the tropes, styles, and themes that the entire trilogy used continue to shape VR storytelling to this day.

During the 1990s, it seemed that everywhere you looked the concept of a totally encompassing virtual reality was a narrative staple in almost every type of tale.  Even an entire episode of Murder She Wrote, was devoted to it as a vital clue to solving a crime.  And it isn’t hard to see why VR as a story element showed up in so many places; the concept was flashy and popular and the alternate reality it represents is, at its heart, exactly what storytelling is about – exploring the ‘what ifs’ and ‘what may have beens’ of life.

But the actualization of VR in real life has fallen far short from the promise of these tales.  Like stories involving space travel and artificial intelligence before it, it is far easier to imagine a far-off future of our hopes than it is to realize the more modest goals that we can achieve.  All of these are instances of intoxication fantasies where the mind constructs an idealistic and totally unachievable future by completely disregarding cold, hard facts.

Fortunately, there are realists who are quite willing to redirect ideas and effort to a future that is not only achievable but actually useful.  Space travel doesn’t take us boldly where no man has gone before but it does let us examine land usage, water pollution, and habitat change from a global vantage point.  It connects distant points on the globe, helps us to remain found (rather than being lost), and make instantaneous communications a reality.  AI no longer means machine sentience with all the dreams of utopia and all sinister overtones of dystopia but rather smart agents that act in specific circumstances and within a well-defined realm of applicability faster and more reliably than can a human.  Modern AI techniques help us to make smarter use of limited resources, connect disparate things together to make informed decisions, and help us manage the ever-increasing scale of modern technological life.

Likewise, it seems that VR is moving away from the pipe dream of total immersion and towards a future where the digital supplements the actual.  A panel I attended at Dragon Con a few years back presented an excellent application of augmented reality.  Imagine you are a network engineer and there is a problem with an onsite telecommunications system of one of your customers.   Despite your expertise, qualifications, and certifications, if you’ve never seen the client’s telecom closet, which may look a lot like this,

it will take you some time just to grasp where each blue cable is going to or coming from before you can even begin to troubleshoot which ones are at the root of the problem.  The network topology may be correctly documented on the appropriate diagrams but wading through all the details, trying to match up some lines on a piece of paper to a real world length of cable in order to find the one circuit that you believe is the culprit is a long, tedious process.  Now suppose that instead of a static network diagram, you are able to done VR goggles that overlay portions of the network diagram over top of the image of the real thing that is in front of your eyes.  Suddenly you have a live, heads-up display to assist your navigation of the network design that rivals the best tutorials that videogames have to offer.  This use of VR is so promising that Koch Industries is touting it in their newest press release.

There are other equally humdrum but interesting applications.  One that is particularly near and dear to my heart is the use of VR to explore complicated 3D data.  I had a recent occasion to look at the tangled magnetic field patterns that arise in certain geophysical situations, and a few minutes in VR space did a lot more for my understanding than months of looking at 2D slices and projections.

So, maybe there is a future after all for VR, just not the one that futurists and science fiction writers envisioned.  Never would have seen that coming.

This month’s installment will be the final column of this series on the chaos game.  As befitting any swan song, this ending should be artistic and dramatic and awe-inspiring.  Unfortunately, the chaos game isn’t really dramatic or awe-inspiring – at least for most people – but it can be quite artistic.  The patterns that can be produced can be quite beautiful and pleasing to the eye.  So, this article will be mostly a tour of our local digital museum displaying the various works of art that one can produce using either the code heretofore presented or with some minor modifications.

The first entry in our tour is the Barnsley fern, which is a self-similar structure invented by Michael Barnsley to resemble the black spleenwort fern.  There are four entries into the set affine mappings that comprise the game:

barnsley_fern = np.array([[0.01,   0,      0,    0,0.16,0,   0],
                          [0.86, 0.85,  0.04,-0.04,0.85,0,1.60],
                          [0.93, 0.20, -0.26, 0.23,0.22,0,1.60],
                          [1.00,-0.15,  0.28, 0.26,0.24,0,0.44]])

When run through the chaos game, a remarkably pleasing fern frond is found (alliteration, a sure sign of class).

Surprisingly, one of the affine mappings in the table has only a 1 percent chance of being selected. Modification of the first entry in the table of affine mappings from 0.01 to 0.2,

barnsley_fern_2 = np.array([[0.20,   0,      0,    0,0.16,0,   0],
                            [0.86, 0.85,  0.04,-0.04,0.85,0,1.60],
                            [0.93, 0.20, -0.26, 0.23,0.22,0,1.60],
                            [1.00,-0.15,  0.28, 0.26,0.24,0,0.44]])

thus increasing the probability of using the first mapping at the expense of the second, results in a slightly less luxurious, or mangier but, nonetheless, quite similar frond.

This result strongly supports the conclusion from the last post that the chaos game operations on this set of affine maps seems to be quite stable against changes.  Play of this kind also opens the door to the idea of a meta chaos game where the exact set of affine maps used at any iteration is either subjected to small random variations or is picked from a larger set.  In the latter case, one can imagine a meta algorithm selecting between the barnsley_fern and barnsley_fern_2 tables for each frond as part of a larger plant or switching between the two at random.  Perhaps this type of numerical experiment will form the subject of a future column.

But now back to the tour.

Other mathematicians, professional and amateur alike, have played with the set of affine mappings to create similar plant-like results.  An interesting example, called here the Flatter Fern, has as its set of affine mappings

flatter_fern = np.array([[0.02,0,0,0,0.25,0,-0.4],
                         [0.86,0.95,0.005,-0.005,0.93,-0.002,0.5],
                         [0.93,0.035,-0.2,0.16,0.04,-0.09,0.02],
                         [1.00,-0.04,0.2,0.16,0.04,0.083,0.12]])

Using this table in the chaos game gives

As in the previous case, the structure is self-similar but each of the leaf groups is thinner and straighter than in the Barnsley fern.

Keeping with the botanical theme, another fan favorite is the Fractal Tree.  Its set of affine maps is

fractal_tree  = np.array([[0.05,   0,    0,    0, 0.5, 0,  0],
                          [0.45,0.42,-0.42, 0.42,0.42, 0,0.2],
                          [0.85,0.42, 0.42,-0.42,0.42, 0,0.2],
                          [1.00, 0.1,    0,    0, 0.1, 0,0.2]])

Running the chaos game with this table gives something that looks more like broccoli than it does a tree.

Once again, one can play with the stability of the game by adjusting the relative probabilities.  By upping the probability of the first map of the set from 0.05 to 0.25, giving the following table

fractal_puff  = np.array([[0.25,   0,    0,    0, 0.5, 0,  0],
 [0.45,0.42,-0.42, 0.42,0.42, 0,0.2],

[0.85,0.42, 0.42,-0.42,0.42, 0,0.2],

[1.00, 0.1,    0,    0, 0.1, 0,0.2]])

the chaos game produces a Fractal Puff

which suggests that the outer edge is soft, fuzzy, and worn away.

In the next wing of the museum awaits a curious set of geometric shapes.  The one we will examine in detail is the Square, whose set of affine maps is displayed in the following table.

square  = np.array([[0.25,0.5,0,0,0.5, 1, 1],
                    [0.50,0.5,0,0,0.5,50, 1],
                    [0.75,0.5,0,0,0.5, 1,50],
                    [1.00,0.5,0,0,0.5,50,50]])

In this case, running the chaos game with this table results in a uniformly filled region.

The trick to making a fractal appear is to restrict the selection of the vertex so that the same vertex cannot be picked in a row.  The easiest modification to the code used in these explorations is to create a new transformation function

def Transform2(point,table,r,prev_N):
    x0 = point[0]
    y0 = point[1]
    for i in range(len(table)):
        if r <= table[i,0]:
            N = i
            break
    if N != prev_N:
        x = table[N,1]*x0 + table[N,2]*y0 + table[N,5]
        y = table[N,3]*x0 + table[N,4]*y0 + table[N,6]
    else:
        x = np.NaN
        y = np.NaN
    
    return np.array([x,y]), N

that is aware of the previous vertex (the variable prev_N) and returns a null result for the new point.   Testing the return value keeps the ‘bad’ point out of the results giving the following pattern

The Wikipedia article on the chaos game, has a stunning gallery of geometric shapes that result from by similar types of rules restricting vertices.  Particularly interesting are the pentagon examples by Edward Haas.

The first one, shows the resulting pattern by using an analogous set of maps to the square and restricting the vertex to be strictly different from the one before.

In Haas’s second case, the pattern results from using the same table as his first case, but with the restriction that the new vertex cannot be 1 or 4 places away from the two previously chosen vertices.

While both cases exhibit five-fold symmetry the differences that arise solely due to the restriction on the allowed vertices is startling.

The final exhibit is a mix of the botanical and the geometric.  The Fractal Tree table was used with the vertex restriction rule used with the square.  The resulting Fractal Twig is beautiful in its brutal desolation (always wanted to talk like a snooty, pretentious modern art dealer)

So, it seems as if the sky is the limit in creating digital art using the chaos game.  I suspect we’ve only scratched the surface.