Transapient Musings of an S6 Archailect Hey there, my name is Bryan Bishop. Here's to trying to keep up with yourself. RSS.
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Transapient Musings of an S6 Archailect
Metacognitive trivialities over smooth topologies and Julian knots of subgeometric spaces; a.k.a mastermind Singularitarian, node of the Larger Submind and Clone of the Ineffable Original.
Even in this golden age of Big Internet Companies Acquiring Everything In Sight, there are still a few independent, small webapps out there that don't get the attention they should for their useful functionality. Every once in a while we get tired of writing about the same big name webapps everyone knows about, so today we're turning the spotlight away from the obvious greats and taking a look at online tools that may have flown under your radar. Here are our top 10 picks of of underhyped webapps that should get more attention than they do. Photo by AP.
Note: There are hundreds of little independent webapps out there, and while we've tried many of them at least once, this list is slanted toward the products I use personally because they filled a need I (or someone close to me) had. Your list might look a lot different than this one—so tell us what we missed in the comments!
Social network creator Build your own social networking community site with Ning, a WYSIWYG web site creator that offers user registration and profiles, polls, blogs, photo albums, and boards at your own URL for free. While at least one LH reader complained about Ning being slow to load for highly-customized communities, it's still mind-blowing that Ning's made creating a dynamic, user-driven web application as easy as snapping together (virtual) Lego pieces.
Voice to text reminders, blog posts, and calendar events If you ever call your answering machine to leave yourself a reminder message, Jott's for you. Call Jott to leave your message instead, and get it transcribed and sent to your email address. You can also call Jott and send your voice message to your blog, Twitter account, or add events to your Google Calendar via Jott as well as other services.
Music library hosting and web playback Stop toting your iPod or entire iTunes library around on your local hard drive: instead, upload unlimited MP3's to Anywhere.FM and listen from your web browser from anywhere. At some point Anywhere.FM may start charging for storage and playback, but while it's free, it's a convenient way to get your tunes on the go.
Group polling for scheduling large events When organizing a recent reunion event that involved upwards of 80 people trying to figure out what the best date was for everyone, Doodle's group polling app saved the day. Set up a Doodle poll with possible dates and times for any kind of gathering with lots of attendees, and email out the Doodle URL, where each person can enter what times they can make it, and see who else is available when.
Browser start page When Lifehacker alum Rick Broida ran his start page showdown, Netvibes was the clear winner. Set up your web dashboard at Netvibes, a sleek, drag and drop, customizable start page that gets all your crucial information in one place the moment you launch your browser. Here are a few ways to trick out your Netvibes.
Web-based office suite While Google Docs seems to be the first product people think of when online office suites come up, the lesser-known Zoho Suite offers more apps and features. See how Zoho stacks up against Google Docs.
Web-based instant messenger Chat with your buddies across multiple IM services without installing a thing using the Meebo web-based chat client. Great for folks in IT lockdown, Meebo lets you log in and chat with buddies on AIM, MSN, Yahoo! Messenger, ICQ, and Jabber/GTalk.
Personal, one-page wiki Even though it's been around for years, we still have a huge crush on the thumb-drive friendly, magical Javascript personal wiki TiddlyWiki. While TW isn't a hosted webapp like the others, it uses a web page to store your information locally. TiddlyWiki has to be seen to be believed, and it's also spawned many offshoots like GTDTiddlyWiki. See how to get organized with GTD TiddlyWiki.
Task manager, reminder system, personal organizer Among the slew of web-based task managers that have flooded the internet in the past couple of years, Remember the Milk stands head and shoulders above the rest: its modern interface and deep feature set make organizing your to-do's actually fun. Here's how to organize your life with Remember the Milk.
Which of your favorite webapps don't get the attention they deserve? Tell us about 'em in the comments.
CTO Geordie Rose revealing his 16 qubit machine in Feb, 2007. A 28 qubit machine was revealed Nov, 2007. There is controversy over the proof that the computers are using quantumness.
Two Falcon Hypersonic Test Vehicles, built by Lockheed Martin with input from NASA and the Defense Advanced Research Projects Agency (Darpa), will take to the air in 2008.
$10 million prize will go to the first group that can sequence 100 genomes (to at least 98 percent coverage and with less than one error per 100,000 bases) in 10 days, for under $10,000 per genome.
Funniest combination of hype and underplay, from rival town TV: X-Prize Draws Hundreds To Las Cruces - KFOX-TV El Paso (Unfortunately, the first day video Vanessa Reyes reports: The X-Prize Contest is no longer active; maybe the fact-checking department found out those rockets didn’t really go to space!) [edit: 10/29 — video is now working: rocketized Learjets raced through the sky navigating a virtual racetrack and Tripoli rockets soared to space — (not really)]
News feed aggregator tiinker aims to help those feeling overwhelmed by their news portals and their own feed readers chop down their lists. Using Digg-style "thumbs up" and "thumbs down" controls, tiinker "learns" what types of news and sources you find relevant and interesting and mixes its content for you accordingly. Tiinker's technology page reshaped itself pretty quickly to a little test-voting with an eye for Linux items, but what's missing right now (and used to be offered) is a way to bring in RSS feeds not offered by the tiinker mix. Still, for those seeking out a decent, customized news portal, Tiinker might be a good way to keep on top of what's happening.
A lot of our makers send in camping gear and equipment, while they're not "MAKEs" there seems to be a lot of makers who camp - so when I saw this wearable sleeping bag I couldn't help but put it on the blog, it's awesome! - [via] Link.
Lachlan Hunt writes "Today W3C announced that the HTML Working Group has published the first public working draft of HTML 5 — A vocabulary and associated APIs for HTML and XHTML. It's been over 9 months since the working group began in March 2007 and this long awaited milestone has finally been achieved. '"HTML is of course a very important standard," said Tim Berners-Lee, author of the first version of HTML and W3C Director. "I am glad to see that the community of developers, including browser vendors, is working together to create the best possible path for the Web..." Some of the most interesting new features for authors are APIs for drawing two-dimensional graphics, embedding and controlling audio and video content, maintaining persistent client-side data storage, and for enabling users to edit documents and parts of documents interactively.' An updated draft of HTML 5 differences from HTML 4 has also been published to help guide you through the changes."
Mac OS X only: One very annoying aspect of status update applications—from Twitter to Facebook to IM—is that there are so many places to update your status, and if you're not diligent you'll end up "out to lunch" for weeks on end. Freeware application MoodBlast updates your status on chat apps, Facebook, Twitter, and more through one universal interface. That means you can enter "Back from lunch" in one place and update the status on nearly every application capable of displaying status updates. MoodBlast is accessible through a universal hotkey, can display the music you're currently listening to in iTunes, your current weather, or whatever individual status you feel like setting. MoodBlast is freeware, Mac OS X only.
What happens when YouTube goes the way of the dinosaur? Slashdot comment -- basically, a distributed video p2p filesharing network is needed to insure redundancy and data integrity, this was the original purpose of the internet, but we've been seeing poorly implemented architecture for over a decade.
A columnist for Wired has an interesting look at how telecommunications are actually making it more interesting to reside in populated areas instead of allowing the complete disregard for distance. "Technology makes it more fun and more profitable to live and work close to the people who matter most to your life and work. Harvard economist Ed Glaeser, an expert on city economies, argues that communications technology and face-to-face interactions are complements like salt and pepper, rather than substitutes like butter and margarine. Paradoxically, your cell phone, email, and Facebook networks are making it more attractive to meet people in the flesh."
Webapp Pingie takes any RSS feed and alerts you of new items in that feed via SMS message. It would be a bit ridiculous to subscribe to the full Lifehacker feed (or other frequently updating sites) with Pingie, because no one really wants 18-plus text messages a day. However, now that you know how to slice and dice Lifehacker feeds to get only the content you want, services like Pingie or previously mentioned Web-alerts could come in very handy with our Feature feed or our daily download recommendations. Pingie is free to use, requires an email address to set up your account.
ScienceDaily (Dec. 18, 2007) — For the first time, scientists at Weill Cornell Medical College in New York City have observed in real time a cellular mechanism that's crucial to how brain cells communicate. In doing so, they've also laid to rest a competing theory as to how key cellular processes—called endocytosis and exocytosis—work.
Healthy neurological function hinges on the efficient passage of information between brain cells via the synapse, and exocytosis/endocytosis is the complex trafficking mechanism that allows this to happen.
At its simplest level, exocytosis involves the packaging, transport and delivery of neurotransmitter chemicals in sac-like structures called vesicles. These vesicles move from the interior of the cell to the cell membrane, where they deliver their information-rich cargo to the synapse. Endocytosis involves a similar function in the reverse direction, with incoming vesicles being transported into the cell's interior.
The vesicles aren't discarded, however. Instead, once they release their cargo they are recycled for use in another go-round. There have been two competing theories about how that recycling occurs—either the vesicle fragments upon delivering its cargo and must be rebuilt, or it simply empties itself like milk from a bottle which is then resealed.
"The vast bulk of the evidence suggests the former theory is actually the correct one, but it's been tempting to think of the 'resealable spout' theory, because it seems so logical and because there's been some ambiguous evidence that it might be true," says the study's co-author Dr. Timothy Ryan, professor of biochemistry at Weill Cornell Medical College.
The trouble is, no one had ever found a way to observe—accurately and in real time—synaptic vesicle recycling as it occurs.
That has changed with this new paper. "We have taken advantage of recent advances in fluorescent 'tagging' of molecules involved in these cellular processes, as well as new microscopy technologies that give us an incredible new ability to watch all of this, up close and in real-time," says Dr. Ryan.
Specifically, Dr. Ryan used a fluorescent chemical stain called pHluorin and genetically fused it to a vesicular protein called vGlut1. "We've used this fluorescent tagging approach before, but with molecules that can exist on either the outside or the inside of the vesicle," Dr. Ryan notes.
"VGlut1 gives us a much more precise view, since it only inhabits the inside of the vesicle," he adds. "That means that when we see the green fluorescent tag move outside of the vesicle, then the vesicle itself must have ruptured in some way. This gives us a much more accurate picture of the recycling process."
At the same time, the team took advantage of new breakthroughs in optical microscopy that maximize how much of the tag's fluorescent light can be "grabbed" by the microscope. This approach allowed them, for the first time, to follow how individual synaptic molecules are delivered and retrieved from the synaptic surface.
"The result is an accurate view into this hitherto mysterious synaptic phenomenon," Dr. Ryan says.
The "resealable spout" hypothesis of vesicular recycling (also known as the "kiss-and-run" theory) may be the first casualty of this new insight.
"We observed that, although recycling appears to occur within a set but somewhat variable time-frame, it's still using the same mechanism—the vesicle falls apart upon delivering its cargo to the cell membrane, and then enzymes go to work re-building it for the next cycle," Dr. Ryan adds. "I think this real-time observation really closes the door on the 'kiss-and-run' theory of vesicular recycling."
The new technology used in these experiments should bring scientists much more insight into how the synapse works generally, and that could have real implications for our understanding of neurological health and illness, Dr. Ryan says.
"This is all part of the 'shop manual' of neurological function that we are currently putting together, piece by piece," he says. "Discoveries like these are adding new pages to the manual every day, and it's that kind of knowledge that will someday save, extend and improve lives."
The study was co-authored by Dr. J. Balaji, who conducted the work while at Weill Cornell. Dr. Balaji has since moved to the University of California, Los Angeles.
The scientists recently published their findings in the online edition of Proceedings of the National Academy of Sciences (Dec. 18 print edition).
This work was funded by grants from the U.S. National Institutes of Health.
Synaptic vesicles are neurotransmitter-containing, membrane-bound organelles. They are initially generated in the cell bodies of neurons, and are subsequently transported along microtubules to axonal terminals. Depolarization of presynaptic nerve terminals leads to CALCIUM INFLUX into the cell. The calcium influx triggers the vesicles to fuse with the plasma membrane and release their neurotransmitter into the synaptic cleft. The rate at which vesicles can be transported from the soma to axon terminals is not sufficient for continuous neural activity. This lead to the hypothesis of SYNAPTIC VESICLE RECYCLING.
Synaptic vesicles filled with the appropriate neurotransmitters dock at the active zone of the pre-synaptic terminal in preparation for neurotransmitter release. When an action potential arrives at the nerve terminal, voltage-gated calcium (Ca++) channels open, causing a sudden influx of Ca++ ions. This pulse of intracellular Ca++ results in membrane fusion between the pre-synaptic terminal and release-ready vesicles. Two distinct components of neurotransmitter release are caused by the Ca++ influx: 1) A fast phase that is induced rapidly, in approximately 50 microseconds following intracellular calcium buildup, and 2) A slower component that continues for >1 s after an action potential (Barrett & Stevens, 1972). Both parts are dependent on the Ca++ influx.
Although fusion is triggered by the Ca++ influx, synapses do not remain static. Electrophysiological recordings of synapses by Katz et al. (8) demonstrated that synapses actually have a low probability of release. These rare vesicle fusions are spontaneous events of exocytosis that result in miniature postsynaptic currents.
Hallmark advances in our study of neurotransmitter release and vesicle recycling are made using the synapse formed by the calyx of Held (a giant brainstem terminal utilized for sound localization). This is the only synaptic model available that accurately accounts for all properties of vesicle fusion and neurotransmitter release (Meinrenken et al., 9). As its name implies, the calyx of Held envelopes the soma of a postsynaptic neuron like a cup, forming a large nerve terminal appropriate for study. About 500-600 synaptic contacts are made between the calyx terminal and the postsnaptic cell. Due to its size, it is possible to make direct electrophysiological recordings within the calyx terminal to monitor both pre- and postsynaptic processes simultaneously.
A crucial question about the synaptic vesicle pathway is how Ca++ channels are organized with respect to the vesicles. Using the calyx model, Borst and Sakmann (7) applied the buffer EGTA to calcium channels, but vesicle release was only inhibited by 50%. Their results suggest that the majority of vesicles are not directly linked to Ca++ channels, and that multiple Ca++ channels contribute to the total ion influx triggered by an action potential.
Bittner and Kennedy initially proposed the idea of vesicle recycling in 1970 (1). Since then, multiple mechanisms have been demonstrated to take place during recycling. In 1973, Heuser and Reese demonstrated the first method of recycling at nerve terminals, called full-collapse fusion (2). In this method, the synaptic vesicles are completely flattened/incorporated into the plasma membrane. Another method, suggested by Bruno Ceccarelli in 1973, was later termed the “kiss-and-run” method. During kiss-and run, the synaptic vesicles fuse transiently with the terminal membrane and reform into functional vesicles (3).
Full-Collapse Fusion model. Synaptic vesicles fuse with plasma membrane, releasing neurotransmitters. Vesicle membrane is subsequently recovered by clathrin-coated pits and recycled through endosomal compartments. Figure taken from Heuser and Reese, 1973
During full-collapse fusion, the synaptic vesicles are incorporated into the plasma membrane, causing a loss of shape, membrane protein, and lipid of the synaptic vesicle (loss of vesicle identity). Figure taken from Harata et al, 2006
This method of fusion/recycling results in a nearly instantaneous release of neurotransmitter, followed by clathrin-mediated recovery of vesicle membrane. Using electron microscopy (EM) to visualize isolated frog neuromuscular junctions pre and post nerve stimulation, Heuser and Reese showed that depletion of synaptic vesicles following nerve stimulation correlated with an increase in plasma membrane surface area. Prolonged stimulation resulted in appearance of endosomes (cisternae) within the cell, which disappeared after rest as vesicles reappeared in the nerve terminals. To determine the timing of movement through these stages of vesicular recycling, they filled the synaptic cleft with horseradish peroxidase (HRP) and again using EM, looked at HRP localization after nerve stimulation. They found HRP in endocytosed coated vesicles immediately after nerve stimulation, followed by localization to endosomes and finally, an hour after stimulation, in synaptic vesicles. Due to the role of clathrin in this process, the recycling is inherently slow due to the kinetics of clathrin assembly/disassembly ( which, at fastest, are >30 s) (10). Although this method was originally observed in frog neuromuscular junctions, it has since been observed in mammalian central nervous system terminals.
During kiss-and-run, various fusion methods are possible depending on the components of the synaptic vesicle that are retained. Figure taken from Harata et al, 2006
The kiss-and-run hypothesis states that neurotransmitter is released through the transient formation of a fusion pore between the vesicle and cell membrane. This term is used to describe various non-classical methods of fusion/recycling, but to date does not have a universally accepted definition (10). The first direct evidence for the kiss-and-run hypothesis was provided independently by Aravanis, Pyle and Tsien and by Gandhi and Stevens in 2003 (4,5). Both groups used fluorescent markers to track individual vesicles in hippocampal neurons following an electrical stimulation.
Aravanis et al used FM-dye labeling to study single vesicular events in hippocampal neurons. Cultures were bathed in FM1-43 and electrically stimulated, causing the dye to be endocytosed through vesicle recycling. FM1-43 was then removed from the extracellular solution, causing it to wash out from the plasma membrane but remain in the protected endocytosed vesicles. Cultures were then stimulated a second time, causing exocytosis and dye release from the labeled vesicles. Aravanis et al. found that upon fusion only a portion of the dye was released from an individual vesicle, and further, another quick electrical stimulation caused slightly more dye release. However if the latency between the two shocks was more than 23 seconds, further release was not seen. These results supported a mechanism slower than the kiss-and-run model (too fast for dye to leave vesicles), but faster than the complete fusion model seen in the frog NMJ.
Gandhi and Stevens used a pH sensitive GFP, pHluorin, fused to the intravesicular domain of synaptobrevin, to monitor vesicle recycling. This fusion protein, synaptopHluorin, fluoresces at high pHs, such as in the synaptic cleft. At low pHs, such as in synaptic vesicles, fluorescence is decreased. Monitoring the timing of fluorescence they concluded there are three modes of vesicle recycling. The first two are similar involving a vesicle that does not collapse and a fusion pore that remains open for less than a second (kiss-and-run) or 8-21 seconds (compensatory). They termed the third and longest (45 seconds) mechanism the stranded mode. In the stranded model, the vesicle membrane incorporates into plasma membrane, but cannot be endocytosed until another action potential causes an additional necessary increase in calcium.
(a) Fluorescence image of full-collapse fusion process. (b) Fluorescence image of kiss-and-run process. (c) Fluorescence drop for (a), showing loss of one quanta worth of intensity. (d) Fluorescence drop for (b), showing several fractional drops in intensity. (e) Histogram showing amount of intensity lost after 20 action potentials. Black bar is for complete fluorescence loss after one action potential (full-collapse fusion). Figure taken from Aravanis et al, 2003
Based on these studies, a kiss-and-run mechanism seemed to dominate vesicle recycling in the mammalian central nervous system. Recently, however, Granseth et al. demonstrated the classical clathrin-mediated mechanism predominates in the hippocampus (6). Using a similar technique as Gandhi and Stevens, Granseth et al. designed sypHy by fusing pHluorin to synaptophysin, another vesicle membrane protein. Using sypHy they found a time constant of 15 seconds for endocytosis. To determine if clathrin is necessary for vesicle recycling, they disrupted the classical pathway by two methods, RNAi knockdown of the clathrin heavy chain and overexpression of a dominant-negative form of the clathrin-adaptor protein, AP180. The resulting complete blockage of endocytosis supported their hypothesis that the clathrin mediated mechanism predominately functions in hippocampal cultures.
Currently, the most convincing argument is that the both methods are utilized to different degrees depending on the stimulation frequency. While kiss-and-run recycling is utilized for high-frequency signals due to the quick recycle rate, full-collapse fusion is better suited for low-frequency signals. Therefore, in order to maintain signaling flexibility, a combination of the two methods provides a broad frequency response that can be sustained over longer time intervals (10). This ability for both mechanisms to be active was demonstrated by Aravnis et al using fluorescent marker FM1-43 to examine rat hippocampal neurons (4).
1. Bittner,G.D. & Kennedy,D. Quantitative aspects of transmitter release. J. Cell Biol. 47, 585-592 (1970).
2. Heuser,J.E. & Reese,T.S. Evidence for recycling of synaptic vesicle membrane during transmitter release at the frog neuromuscular junction. J. Cell Biol. 57, 315-344 (1973).
3. Ceccarelli,B., Hurlbut,W.P. & Mauro,A. Turnover of transmitter and synaptic vesicles at the frog neuromuscular junction. J. Cell Biol. 57, 499-524 (1973).
4. Aravanis,A.M., Pyle,J.L. & Tsien,R.W. Single synaptic vesicles fusing transiently and successively without loss of identity. Nature 423, 643-647 (2003).
5. Gandhi,S.P. & Stevens,C.F. Three modes of synaptic vesicular recycling revealed by single-vesicle imaging. Nature 423, 607-613 (2003).
6. Granseth,B., Odermatt,B., Royle,S.J. & Lagnado,L. Clathrin-mediated endocytosis is the dominant mechanism of vesicle retrieval at hippocampal synapses. Neuron 51, 773-786 (2006).
7. Borst J, Sakmann B. Calcium influx and transmitter release in a fast CNS synapse. Nature 383, 431-34 (1996).
8. Katz B. The Release of Neural Transmitter Substances. Liverpool: Liverpool Univ. Press (1969).
9. Meinrenken CJ, Borst J, Sakmann B. Calcium secretion coupling at calyx of Held governed by non-uniform channel-vesicle topography. J. Neurosci. 22, 1648-67 (2002).
10. Harata NC, Aravanis AM, Tsien RW. Kiss-and-run and full-collapse fusion as modes of exo-endocytosis in neurosecretion. J. Neurochem. 97, 1546–1570 (2006).
If tracking your calorie intake is part of your New Year's resolution and you already use Twitter, create and maintain a food diary using the Twitter-enabled Tweetwhatyoueat service. Here's how it works: on Twitter, friend the Tweetwhatyoueat bot (named twye) and direct message it your meals and snacks throughout the day from the web, your phone, instant messenger, or your Twitter client of choice. Tweetwhatyoueat creates a food diary based on your direct messages, calculating your calorie totals as you go. (Your food intake isn't public, so your friends won't know about those Twinkies.) Being a closet Twitter fan, I love this new way to make the service actually productive. After the jump, check out Tweetwhatyoueat in action.
Here's what my food diary from this morning looks like:
It took a few tries to get Tweetwhatyoueat to friend me and start my diary, and it's also not that smart figuring out how many calories your food items contain itself. But you can go to your diary and enter calorie counts by hand at the end of the day, and armed with a site like aCalorieCounter.com, that's no big deal. If you know the calorie count of what you're eating off the top, you can specify it in your direct message, i.e., messaging it like so: d twye 1 small orange:100 will enter "1 small orange" into your food diary, with a calorie count of 100. Overall, Tweetwhatyoueat is a clever way to put Twitter's platform to good use.
Windows only: Free copying utility Text Mining Tool grabs just the text out of Word documents, PDFs, HTML pages, and other documents without the hassle of opening, selecting everything and hoping embedded images don't leave strange markers in the text. Once your text is copied, you can either re-save it as a text file or copy it to the clipboard. Its function might not sound all that helpful—until you've tried to select multiple pages' worth of text from a scanned PDF, or tried to grab text from around awkward Flash boxes on web sites. Text Mining Tool unzips to a folder that can be put anywhere and comes with a command line tool for your batch-script-writing pleasure. Text Mining Tool is a free download for Windows systems only. For similar copy power from the selection screen, try DragKing.
Working with the command line can save time, but most of us non-programmers know only a few key commands to use. Using the apropos command, however, anyone can search for commands and programs that relate to whatever keyword you search for. If, for instance, you knew you had a video encoder handy but didn't quite know how to get at it, type in apropos mpeg and you'll get a list of commands and programs that have the words "mpeg" in their man, or manual, files. Great tip for beginners and terminal hackers alike, and apropos is installed on a wide number of Linux distros and other terminals. For more beginner help, check out Unix resources for newbies.
The "Walking Radar" project connects up a Basic Stamp and IR sensor to your cell phone in order to detect objects in the environment and influence games on the device or trigger SMS messages to be sent. Pretty cool, just don't try to take this phone through airport security.
Firefox with Greasmonkey: If you get carried away surfing the web by clicking every tangential link in sight, drawing you down yet another road away from the stuff you're supposed to be doing, check out the No Links Please! Greasemonkey script. This simple user script strips web pages of links, ensuring that you stay where you are, get the info you need, and get done. Its creator explains:
One of the things which makes the web great are its hyperlinks. However, they also make the web vast and most importantly, far too easy to roam. No Links Please! breaks the web by removing hyperlinks from all pages apart from Google. Without the knowledge or temptation of links you are free to devote all your time to real work and never roam the web again.
Obviously the web isn't the web without links, so you could apply No Links Please! to individual sites (like Wikipedia, for example) to save yourself from particular hyperlink distraction sinkholes. No Links Please! is a free download and works with the Greasemonkey Firefox extension.
Over at 43 Folders, writer Brian Oberkirch looks at how one of his favorite artists beats his worst tendencies to be intimidated by the hugeness of the work in front of him. By dividing his work into a grid and breaking it into smaller incremental tasks—like knitting a sweater—he sidesteps his anxiety.
What I found that one of the nice things [about] working incrementally is that I don't have to reinvent the wheel every single day. Today I did what I did. You can pick it up and put it down. I don't have to wait for inspiration. There are no good days or bad days. Every day essentially builds positively on what I did the day before.
Basically, we consider each possible path from the source — which we’ll assume is at the origin of our lattice — to the target — which will be at lattice coordinates — as a series circuit, and the collection of all possible paths is a giant parallel circuit. If a given path has length it passes through resistors, and so its total resistance is . Thus we need to form the sum over all paths
and take the reciprocal of this sum. So we’ve got a combinatorial problem: how many lattice paths have length and go from the origin to ? Let’s call this number and build the “exponential generating function”
We choose the exponential generating function here rather than the ordinary generating function because of a manipulation we want to do later that looks nicer for exponential generating functions. We’ll go from to our desired sum a little later.
For now, let’s consider a simpler generating function. Consider the number of paths on a one-dimensional lattice with length that start at and end at . First off, if we have no paths that work. Similarly, if is not divisible by we won’t have any good paths. But when , there are generally many different paths. How do we get our hands on one? Choose steps to take left and the other to take right, and we’ll end up steps to the right. So we set to get the exponential generating function
where is a Bessel function of the first kind.
Now a path in the two-dimensional lattice is really a mixture of two one-dimensional paths. That is, we break our steps up into chunks of and steps, respectively, put a one-dimensional path on the chunk ending at , and put a one-dimensional path on the chunk ending at . And it turns out that generating functions are perfect for handling this!
This tells us how to find the number of paths of length that end at — it’s just the th derivative of this power series, evaluated at . That is
Now let’s consider the ordinary generating function for :
Here the coefficient of is the number of paths of length (and thus with resistance ) from the origin to . What we want to calculate is the sum . To get at this, we’ll divide our power series by and then integrate with respect to . This will give the series , which we can then evaluate at .