When Sarah gets the hiccups, she really gets the hiccups. A special little holiday edition for those of you who know her. And for those of you who don't, I don't even know what to say. :-D
Sarah's hiccups (260 K wav file)
DiveIntoMark is a continual source of fun anecdotes to show Sarah. He wins points for a good, sometimes self-deprecating, sense of humor, quotes from Winnie ther Pooh, and recent talk about babies. I like the added mix of zork, math and general geekiness. Today Mark has posted a story about the Cantor set, Sierpinski carpet, and Menger sponge. Sarah will not be amused, but I am. The ensuing discussion about various shapes of infinity has prompted me to publish the Vanishing Point animation I have been working on for the past several months.
When I was in fifth grade (age 10ish) there was a wooden cone at school sliced into three pieces that revealed three conic sections: circle, ellipse, and parabola. At the time I was curious to know what the relationship was between the slope of the intersecting plane relative to the axis of the cone and the resulting conic section. I didn't have the vocabulary to articulate it that way at the time, but that's what I wanted to know.
I have always loved geometry (fun and interesting) and tolerated algebra (boring). On the high-school geometry final exam (age 15ish), the instructor informed us that the final proof on the test required at least fourteen steps -- if you had less you had it wrong. I solved the proof in seven steps. I had finished the test about twenty minutes early and had plenty of time to double-check my work. I finally asked the instructor to go over my solution and he agreed that my solution was correct.
Fast forward to college. I enrolled in Calculus III as an elective even though it wasn't part of the architectural requirements. I was specificall looking forward to formally answering that question about cones and intersecting planes (finally). They were using the same text book as Calc II and the next couple chapters covered three-dimensional geometry.
One of the first things out of the instructor's mouth was "We're going to skip the chapters on three-dimensional geometry because it really isn't very interesting. Instead we'll jump straight into infinite series."
Bastard.
I tried to study those chapters on my own and couldn't quickly figure out how to apply the knowledge to answer my question. While I was trying, I managed to answer the question by inspection and gave up on the calculations.
There are two lines of interest: the axis and the edge of the cone. A circle is easy. The intersecting plane must be perpendicular to the axis of the cone. An ellipse is a generalization of a circle. The plane must intersect both edges of the cone. To create a parabola, the intersecting plane must be parallel to one of the edges. And for a hyperbola the plane must be somewhere in the range between the edges of the cone. I'm ignoring some edge cases (pun is intentional). Planes which intersect at the point of the cone generate a point. And Planes which intersect at the tangency generate lines. Perhaps that means that points and lines should be included in the family of conic sections. :-)
(yet another addition to my growing list of things to animate :)
This is a reconstruction of the animation that led to five months of working with Dr. Kim Veltman in Italy and some of the richest learning experiences of my life.
In linear perspective, when parallel lines are projected onto a picture plane [1], the resulting projections [2] will intersect at a vanishing point. For lines parallel to the ground, the vanishing points will appear on the horizon line [3]. When drawing objects in perspective, the artist usually begins by drawing the horizon line and a couple vanishing points. Construction lines are drawn from those vanishing points to help the artist correctly locate the edges of buildings or other objects. Perspective drawings and paintings are sometimes classified by the number of vanishing points used in the construction.
I don't remember the exact definition of a vanishing point from Dr. Veltman's collection. It would have been similar to what I've written above: accurate and descriptive with a certain amount of precision in the use of language and inherently relative to other terms of perspective. But words don't suffice. A vanishing point demands illustration. I found my original diagram of a vanishing point painfully lacking. It looked basically like a stick figure and a pile of triangles. At the time I was very interested in animation and thought I could visualize a vanishing point more clearly with a little motion. I ended up with a stick figure and a moving pile of triangles which was much better. ;-)
Without further adieu (with the caveat that an SVG plugin is required) . . . drumroll . . . (click Go):
It is probably not self evident as to why Kim was so impressed by such a simple animation. Computers as an Historical Tool for Mathematics, Science and Art will help clarify why this is an interesting contribution to the history of perspective. There are some other hints in another article which I quote below.
Back to the pictures of infinity. Notice how as the projected lines grow ever more slowly towards the vanishing point before leaping at the very end. This is a direct consequence of geometry and infinity. The ground lines are growing at a constant rate whereas their projections are growing in a ratio relative to the point of view and position of the picture plane. If I let it keep going it does eventually get fairly close to the end, but it's really not worth the wait. Someday I'll rearrange the animation so the projected lines grow at a constant rate. I'll still have to do some fudging towards the end as the ground lines approach the limits of the SVG coordinate system.
Speaking of coordinate systems there's another detail worth mentioning if you're inclined to look at the code behind the animation. SVG uses a two-dimensional coordinate system whereas the scene I'm depicting is three-dimensional. For this animation I'm just using the 2D coordinates to calculate intersections. In the AutoCAD version I had the advantage of a three dimensional coordinate system for my code. That arrangement more accurately reflects the actual geometry at work when light reflects off of objects and into our eyes. For simplicity my stick figure is a cyclops. In both of these ways I'm actually continuing a very long tradition in the history of perspective and geometry in general. :-)
Here's how Kim described it in the introduction to Computers and Renaissance Perspective
A long tradition of Euclidean geometry developed two-dimensional conventions of representation to the extent that they were part of the legitimation process in mathematics. As a result, Renaissance treatises on perspective evidence a basic paradox: they use abstract two-dimensional conventions to display the principles of a new three-dimensional method of representing space. This is achieved by folding different planes (usually a lateral view and or a ground view) into a single plane (usually a frontal view). This procedure of folding back (technically termed ribaltimento in Italian and rabattement in French), makes most of the diagrams in the early treatises virtually incomprehensible to the untrained eye, all the more so because the reader is confronted with a completed construction which usually gives no visual clues concerning the steps taken to get there. One can identify the steps taken in arriving at an end product in any of these constructions; one can reconstruct these steps and theoretically it would be possible to print these, except that the cost of including so many diagrams makes this alternative prohibitively expensive. All of which helps explain why these treatises have never been studied systematically.
Many of the animations I created in Siena were folding the two-dimensional images into their three-dimensional origins. In one, I animated the steps Piero della Francesca described to construct a perspective image of a pentagon. No small task to understand a centuries old Italian text on a geometric construction. When I was done animating his instructions I re-did the animation in a way that I thought made the technique he was describing more clear. If I remember correctly, I then folded the resulting drawing into its three-dimensional arrangement and animated something of a proof of the technique by projecting lines from the point of view to the pentagon on the ground and showing that constructed image aligns correctly. When I get more tuits I'll see if I can repeat that stunt. :-)
[1,2] See my illustration of perspective basics
[3] I still haven't illustrated a horizon line anywhere.
Over the weekend Sarah and I rented Winged Migration. Wings and migration are themes in my dream encyclopedia. The film feels like a step in the direction of what I was imagining. I was reminded of my early correspondence with Kim Veltman when I first articulated that dream.
I awoke early this morning and began catching up on my bloghopping. Seb bought my attention to an article by Valdis Krebs : What's Your Google Number?. To get a sense of it I googled for "Eric Dobbs" AND Veltman and found myself re-reading some of Kim's articles.
Veltman's got an ambitious goal: the restructuring of the whole of human knowledge. He's entirely serious about it.
From How it all began
As I reflected on the magnitude of what I had experienced I was overwhelmed by a sense of sadness and near despair. I had just had an incredible trip. I had seen more culture and civilization in three months than many persons in a lifetime. I had seen the great centres of Greek, Roman, and Hittite civilization. They were all in ruins. It wasn't as if there were just one or two sites that had fallen into a slight state of disrepair. They were ruins in the truest sense. My sadness and despair came from thinking: if this is how civilization treats its highlights, the best it has to offer, what hope is there for civilization?
Spirit
It took me a decade to find a tentative answer. The highlights of civilization which I had seen, the temples at Selinunte, Agrigento, Ephesus; the colosseums at Arles and El-Djem; the theatres at Epidaurus, Segesta, Taormina and Aspendos were all physical manifestations of social customs and spiritual ideals. There is the spirit and the flesh. There is the human spirit and there are objects, which are expressions of this spirit. As the spirit evolves, the expressions change and the earlier expressions are reduced to being merely objects and are thus neglected and forgotten. The Roman custom of combat made colosseums necessary. But there was something fairly primitive in a custom that included feeding Christians and slaves to lions. So when the customs improved, the buildings of the old customs were neglected. Ultimately they became ruins because the ideas, the ideals, the spirit behind them was no more. And while they continue to have an historical value in reminding us of what once was, the real challenge lies elsewhere, focussing on the spirit rather than its expression, the soul of culture not its skin, the dreams. If the dreams are right, if the spirit has a true vision then the buildings, the monuments will follow. The monuments are not the civilization. They are merely expressions of a spirit and will crumble the moment the spirit moves on. Gradually the quest became to find the spirit of civilization, or rather to cultivate a spirit that could lead civilization in new directions.
....
Learning is one of the most fascinating aspects of culture and civilization. There is inevitably a tendency to learn about ourselves but not about others. Anyone who was not a Greek was a barbarian. The Romans built Hadrian’s wall to keep the so-called barbarians from Scotland at bay. The Chinese built a wall around their culture for similar reasons. Yet the memorable civilizations have been precisely those which transcended that limitation. Aristotle became one of the greatest persons of all time because he commissioned his best student, Alexander the Great, to help him learn about things in Persia, India and wherever he went. The library at Alexandria tried to collect learning from all known cultures. Arabic culture became great when a caliph at Gundishapur sent emissaries to the west to collect Greek and Roman culture. Today, the Vatican, Herzog August Bibliothek, British Library, Bibliothèque Nationale de la France, and Library of Congress are the greatest libraries of the world precisely because they did not limit themselves to learning only about their own culture.
These thoughts churned as I walked to the bus station. I wondered about the size of the gap between analog and digital knowledge. As a computer geek it's easy to fall under the illusion that everything interesting or relevant can be found with Google and persistence. But I suspect those five libraries hold vastly more knowledge, and vastly more important knowledge than anything Google can scour with its spiders. As we charge toward digital and away from analog knowledge, how much will we leave behind?
Charles Miller blogged about Etherial Memory
I am increasingly finding that the Internet is my swap-space. I've stopped remembering "things", and started keeping a catalogue of references to things. If I find I need the information, I have enough recollection of where to find it that I can access the Internet and swap it back into main memory. Luckily, I have a really good associative memory for context, which helps me come up with the Google search I need to track down data I've otherwise swapped out.
This sentiment is remarkably similar to lessons I learned by osmosis while working with Veltman. For millennia, scholars have been teaching the next generation of scholars how to navigate libraries and museums and how to document one's travels for future generations.
What we computer geeks are doing is mostly abandoning that legacy and reinventing it in digital form. Libraries are driven by meta-data. There are many, many systems to categorize books. In the US we use Library of Congress or Dewey. There are dictionaries to help one learn the terminology and jargon of different fields. There are indexes of major works. Bibliographies. Footnotes. Secondary literature. Encyclopedia. Meta-data by the ton. Veltman would like to encapsulate the millennia of scholarly meta-data navigation skills into a tool for the common man.
Juxtaposition
These thoughts were turning over in my mind. About a block away I noticed a fire truck turning into the bus station. Moments later I saw the express bus I was hoping to catch leaving the station in an unusual direction. As I approached I could see the fire fighters hosing down the bus's usual exit. The driver of the slow bus to Denver explained that someone had jumped off the top of the parking garage to their death. Thinking about what they were cleaning off the pavement sent a shudder down my spine.
I've been thinking about suicide and terrorists a lot lately. My instincts tell me that our fight against terrorism is completely backwards. Israel offers a continuing demonstration that force is ineffective in stopping enemies who are willing to kill themselves to make their point. I've heard the rumors about how Palestinian suicide bombers are considered martyrs and somehow earn points for their families in heaven. But humans have strong survival instincts. Suicide requires more than just a holy purpose -- in order to override their survival instinct people have to be deeply despaired and more scared of living than they are of dying. Adding violence on top of violence only increases the despair. It would be more effective to find ways to foster hope and reasons to live.