The Path Forward

2023-12-18: "Alright, Mr. DeMille, I'm ready for my close-up."

If ever there was a day when I thought that my website, which deals with an obscure little topic called "Unicursal Maze Research" would have 14 followers or have a single-day total of 192 views... Well, everyone dreams of winning the lottery but, odds are, dreams are the only place where such events tend to happen.

However, when there is an audience in the seats, as the saying goes, the show must go on. And it's not as though that I haven't tried to make this website a bit more interactive. The POINTS learning module, despite it's sudden appearance, didn't simply emerge overnight. Admittedly, though, it was spurred toward completion thanks to the increased viewership. There are at least two other learning modules in production. Perhaps one of them finds it's way onto the website before the New Year? Anything is possible in a world where this website is suddenly receiving more than 20 views per day.

For those who enjoy a little 'insider baseball' gossip, I have been researching ways of allowing people to walk through the very unicursal paths that I have been creating. Yet Neocities is rather restrictive on what can and can not be placed onto their website (as the ".png vs .PNG" fiasco from a short while ago demonstrated). Therefore, resurrecting an old (and, admittedly, proprietary) HTML format called 3DML that I had some familiarity with is no longer a viable option. An absolute pity that it can't be restored here in a place that seems to celebrate a revival of the old Internet spirit.

From javascript implementations of raycasting engines to TrueType fonts to another obscure VRML format, I've been scouring the Internet in an effort to find ways of making my topic a little more exciting and inviting. For every minute that I'm working on website presentation, though, I'm spending one minute less on unicursal maze research itself. It's all well and good to blow the dust off of obscure HTML formats (proprietary or W3C-approved) and remember 'the good ol' days' of 'blink' and 'marquee' HTML tags (and I can remember back when Cascading Style Sheets was still a proposal and not a standard), but an empty website, no matter how pretty, is still an empty website.

Popularity is fleeting. Chasing after hits, views, followers, subscribers and likes can be intoxicating. I'm no different than anyone else. Who doesn't want to walk down a red carpet wearing a tuxedo with numerous flashbulbs going off? It's a lot of hard work to create unicursal mazes. Even with a conservative estimate of one hour of work per day, that's still over two whole weeks per year. That's a lot of time to devote to any one pursuit, trivial or otherwise. Who doesn't want to be recognized for their efforts?

As addictive as the chase for popularity may be, though, I have to keep remembering that the website was born from the research and not the other way around.

It's time to get back to work.

2023-12-03: I have seen the future of unicursal maze research. It is more exhilirating and frightening.

As the determined fields get larger, there will be more days when I'll be processing seven cycles per day, every day. Will I be prompted, through necessity, to break with my recent tradition of only doing a maximum of seven cycles per day if I ever want to see the prospect of processing an Nx8 field? There is a non-zero chance of such a change occurring. But, for now, a maximum of seven cycles per day is plenty for me... While I am gainfully employed and far from the prospect of retirement, of course.

Recent publicity on Neocities has compelled me to work on certain sections of my website. A more advanced practicioner of the HTML arts would make little work of what I currently struggle with. Nevertheless, additional exhibits are being developed. I am even thinking that the exhibits and learning modules (for which there is still only one after all of this time -- No need for reminding me) are really just two sides of the same edutainment coin.

Although the dopamine hit that additional views from other netizens create is intoxicating (and fleeting, based upon the chart that Neocities provides), the real wealth of unicursal maze research is... Well, the research itself. Carving out the actual rules of how to create an unicursal maze besides the brute force method that I currently employ has advanced to the point where I have made this epiphany: I'm not very proficient at writing out clear, well-defined rules. I know the rules well enough but well enough to write them down clearly and hand them to someone else and have them understand them without having to use an additional twenty minutes of explanation and examples? If only I could.

2023-11-11: I keep meaning to write a blog entry and I keep not writing it. I am likely not the first person to realize that my website-maintenance manners (or extreme lack thereof) is perhaps the reason why my daily view numbers are not robust. However, I also understand that the subject material for this website is not enticing to the majority of netizens. Regardless, some form of communication is in order.

First and foremost, those waiting with baited breath to see the end of the first original pair of the 6x6 determined field should consider exhaling and getting about their daily lives. I won't be dramatic and write that there is 'no end in sight.' I can't, though, in good conscience, claim that 'the end is near,' either. I do think that I will be far closer to the end of drawing all of the original mazes for the first original pair in about one month (mid-December). How close to the end will be a lot more accurate than now.

In order to place a lid on this topic, I curently have 742 files for the first original pair alone (those are cycles, which are uncompleted mazes, combined with completed mazes). For comparison, the largest number of files for a 5x6 determined field original pair is 757 files... Needless to write, the number of completed mazes will likely be massive (at least, for me). 500+ original mazes from one original pair alone? That estimate is not entirely without merit at this point.

Speaking of original pairs, I am nearing the end of working on the 6x9 field for determining how many original pairs in that field there are. I just discovered the 160th original pair and estimate about another 20 or 30 to go. Although 7x9 will likely be quicker to find all of the original pairs, 8x9 will be a nearly year-long event. Yes -- YEAR. I highly suspect that I won't complete finding the original pairs for the 8x9 determined field until early 2025 (that wasn't a typo). Mark your calendars?

My research into creating original paths for diagonal fields and undetermined fields is very casual; The star of the research are the determined fields. I put research into the other two types for mere curiosity and to be a 'completist' more than anything else. For diagonals, I am still working my way through the 3x6 field. The undetermined mazes are currently at 5x5.

With analyzing determined paths, I only recently started original pair #9 for the 5x6 field. I will spare anyone reading the suspense: I'm not going to 'catch up' to where I am at with the 6x6 field. The pacing is not designed in that way. Even if, by some miracle, I was to catch up, that type of progress would be bad as I want to allow some time between discovery and analysis to ensure that I haven't missed any obvious paths. Research may be slow but slow yields less mistakes than fast (and, usually, reckless).

I hope that my daily submissions of the latest new path is entertaining. Not every cycle yields a new path as the few who follow the website has observed. However, persistence is the key to all progress and I am making progress. I'll be done with the first original pair of the 6x6 field in the not-so-distant future and there are only 45 in total. Estimating a completion date for the 6x6 at the current rate is more than just speculative and only one or two rungs of the ladder above a 'your guess is as good as mine' admission. However, for the sake for fun, why not write the year 2030?

Time to get back to work.

2023-09-24: I will be the first to admit that I should do more with undetermined paths. Although the "star of the show" are the determined paths, the undetermined paths deserve some attention as well. I sometimes wonder if, in an alternate future, the emphasis of my research would be with the undetermined paths instead of the determined paths if only for a few arbitrary decisions which have since been lost to time.

In one area, I am attempting to rectify the deficit of attention towards undetermined paths and that is in the topic of "looped paths." A looped path, as a reminder, are simply those paths where the beginning and end are right next to each other. This causes the two ends to be easily 'linked' and create an endless 'loop.'

Up until now, looped paths have been derived from determined paths. Not anymore. I have now begun the process of converting applicable undetermined paths into looped paths as well. There have been challenges in this new endeavor but none that have been insurmountable.

Undetermined paths are never going to be equal in comparison to determined paths in terms of attention. That is not to say that there isn't legitimate interest with them. They will always 'lag behind' determined paths. Yet the gap is closing, albeit slowly. All research, conducted properly, is good research and research into undetermined paths needs to occur if new revelations are to ever be discovered.

2023-09-17: The first completed determined 6x6 path has been created. It took 40 cycles to get there.

It's been a long while since I've done so many cycles in a row without creating a completed path. It felt odd to go through that drought but I know that this is going to be the "new normal" when it comes to starting a new original pair. As the fields get larger, the initial time from starting an original pair until you create your first completed path will become longer.

Creating the first path in a new field is something to celebrate, though. It's another milestone with many more to follow.

2023-09-11: The day is here. Years in the making. The final determined 5x6 original path has been drawn.

7,963 paths. I'm not a fool; There could be more. I found a new original path not too long ago, remember? I'm humble enough to admit that there may also be some that I might not ever find.

Tomorrow, though, will be the first day in a long, long while where I don't visit the 5x6 field as 'an active worksite' anymore.

The accomplishment is still sinking in. I'll write more later.

Tomorrow begins a new era. The 6x6 field. It'll probably be days before I generate the first completed original path.

I. CAN'T. WAIT.

2023-09-10: Tomorrow is the day that I complete the determined 5x6 field for original paths. That's it. The End. No more.

To be absolutely clear, I'm not stopping in creating more paths. Next up is the 6x6 field and so on and so forth. I will continue to create new paths for the foreseeable future.

I should be happy. I should be thrilled. And, in some ways, I am. This is a huge achievement for me. I won't be bold enough to say that this is a triumphant day for science or math or anything such as that. I have no doubt that an eighth-grader proficient in programming can type out a program to perform in minutes or hours what I've done in YEARS.

Yet I feel a bit sad over the entire event. I started this particular field with beloved pets who are no longer with me. I started this field with a computer that doggedly, and very reliably, stood by my side through thick and thin while I established my current methods for path generation. Had Microsoft not ended Windows 7 support, I would still be using it today.

There are no guarantees in life; We are at the mercy of decisions not of our own making. I have no idea if I shall see the end of the 6x6 field or not. I know, full well, that I am generating templates for original paths right now that I will likely never start unless I radically increase my productivity. Even starting on the 10x10 field is unlikely, never mind any of the Nx9 fields. At this point, it would be fair to say that, maybe, maybe I might start the 2x8 field in my lifetime if I continue on my current path of productivity. Would I still be alive to see the end of the 8x8 field? Probably not.

It's humbling to write that I don't have any great celebration in store for this momentous occasion. I have other interests and this particular one takes up roughly one hour a day, every day. That's before all of the necessary website maintenance and content creation. I am not very good at presenting my interest.

As an aside, tomorrow's final paths have no direct correlation with the horrible events that occurred on September 11, 2001. It is a mere coincidence that the final paths will be drawn tomorrow and nothing else.

2023-08-26: I found an error today that I had made over three-and-a-half years ago. Finding such errors keeps you humble and forces you to concede that, for as skilled as I might be in creating paths after all of these years, danger always lurks to those who are not attentive at all times.

The only comfort that I may take in this latest lesson in humility was that the error did not occur in what I consider to be "the modern era" where I parse my work more carefully by limiting the number of cycles per day. Of course, such an additional rule means nothing if you are not careful; Errors can occur regardless of the quantity of work that you perform.

As always, I must be grateful to have caught this error, albeit years after the fact. The more accurate the data, the more accurate the analysis. Hopefully, someday, that analysis can yield to methods of error correction that will prevent such errors from occurring in the future.

And, as a final note in an effort to soothe a bruised ego, this incident is another reminder of why I create each path myself rather than relying upon a computer to automatically generate the paths for me. It is this level of experience, gained by creating the paths myself, that allowed me to instantly recognize the error upon seeing it.

2023-08-21: Here we go... The final original pair of the 5x6 field. There are 66 such original pairs; Today, I started working on paths for the final one.

How many days will it take to complete? Who knows. A week? Two? More? The general trend is that the number of paths per original pair decreases with each pair.

The anticipation, though, is building. The end of the 5x6 field. Years in the making. 7,000+ original paths thus far.

I can't wait.

2023-08-15: I am now all caught up from the time that I took off. It took far longer than I expected but, in hindsight, I should have suspected a long haul to occur. "Standard Work," as earlier described, is the most amount of work that I perform in one day before I psychologically consider it "work" as opposed to a "hobby" or a "pastime." That distinction is important because part of the strategy is to perform "Standard Work" every day, regardless if I have a headache or I'm sad or I'm angry or some other circumstance.

Regardless, I'm all caught up now and, every day that you see the latest completed determined path... That's the bleeding edge of progress. And I'm getting excited because there's only one more original pair to go through after this current one before I end the 5x6 field for good. Forever. It's felt like it's taken forever.

Coming soon? The 6x6 field. I can't wait.

2023-04-28: In a few days, I will be taking an extended break from working on Unicural Mazes for social functions. I shall resume my work by mid-May. I apologize for any inconvenience this may cause visitors. I do intend to "make up" for the lost work so that a normal progression of work may resume, as if no break had occurred at all.

In compensation for this extended break, Exhibit #001 in the gallery shall remain open indefinitely until, at least, the extended break ends. The premiere of Exhibit #003 shall be delayed until after the extended break ends. Thank you for your understanding.

2023-04-15: Finding all of the paths in the latest diagonal unicursal maze original pair took me by surprise. I knew that I was reaching the end but I thought that I had more cycles to go. Just goes to show how easily you can lose track of time.

I only perform one cycle per day when it comes to diagonal paths. Slow? Absolutely, but it's not a critical field of research. There's some amusement in researching it and I'm genuinely curious about the results. I'm not losing any sleep if nothing ever comes of it (after all, I'm only performing one cycle per day).

However, the completion of that original pair also reminds me of a terrible milestone: I started that original pair back when one of our family pets was still alive and who, unfortunately, has since passed. And, sadly, this is not the first time that I have started on some milestone when a family pet was alive and passed before that milestone was achieved.

Actions have consequences and time is not forgiving to anyone for any reason. As I complete milestones in my research, I must constantly remind myself that the passage of time will take us all at some point. How far shall I get before it's my turn? As morbid as such thoughts are, I must think of them nonetheless.

And, maybe, that's why I only do so much research per day -- So that I may enjoy the rest of it on other ventures. As I get older, I am trying to 'stop and smell the roses,' so to speak. Sure, I may not achieve as many milestones as I would like by being 'casual' in my research but what would I be sacrificing if I was slavishly dedicated?

2023-03-26: As I have written before, I only perform so much work in Universal Maze Research per day. Only so many cycles performed in determined fields. Only one cycle in undetermined fields and diagonal fields apiece. Only one original pair discovered. I won't describe my entire daily work load as I have described it before but the work is not extensive. The entire day and night session lasts... An hour? A little more than an hour, depending upon the peculiarities of that particular day? It's not a lot.

"Why don't you do more?" is the inevitable question. And the answer is simple: Because I want to do this EVERY day. And if I want to perform this type of work every day, I have to pare that work down to something that is mentally, emotionally & intellectually feasible. I have been to that 'other side of the coin,' if you will, and that side is not all roses and rainbows. Burning yourself out does not bode well for your progress. Therefore, I have to design a regiment of research that I can perform every day: Rain or snow, cloudy or sunny, good times and bad. I have performed research on days that pets have died. I have performed research on days where I have had really bad headaches and have not otherwise felt physically well. I have performed research on my birthday and on holidays. There have been very, VERY few days for, at least, the past five years where I haven't performed daily research... And, even then, I have quickly made up for it on the very next day.

It would be easy, especially on days where I am enthusiastic, to work a lot harder. However, I know that, if I want to go far, I need to take it slow and steady. Today was a tough day, mentally, to perform the daily research. Tomorrow, I might breeze through it without a thought in the world. That's just how this research is. And, inevitably, I will get there, I will complete fields and make discoveries. This research is a marathon, not a sprint. I've learnt that lesson the hard way.

2023-03-22: Before paths can be created, you have to determine what the original pairs are in a given field. I've just finished the determined 04x09 field earlier today. When will I actually work on that field in terms of path creation? I don't know. Honestly... Maybe never. I'm currently still on 05x06 and, after that is 06x06. Then comes all of the Nx7 fields (02x07, 03x07, etc.) followed by the Nx8 fields... There's 16 whole fields -- not original pairs -- FIELDS, that I would have to complete before I could get to the 04x09 field.

I quickly realized, walking into this research years ago, that I would never reach the end. It's not about reaching 'the end' (which doesn't exist) but how far you are willing to commit. But isn't that the case in all endeavors?

2023-03-15: In two days, I'll be completing the 0506-57 (technically 0506-05J) determined field. The original pairs have been going quick lately; 5 original pairs in 2023 alone so far and it's only mid-March. Will I be able to complete the entire 05x06 field by the end of 2023? It's possible. There are 9 more original pairs after this current one. I have no doubt that some of them will take longer than some of the more recent original pairs.

I don't want to look too far ahead but 06x06 keeps getting closer and closer...

2023-03-12: One question people might have in their minds is 'Why don't you do MORE per day? Why only so many cycles of THIS and so many cycles of THAT?' It's a fair question. The truth is that, at this point, I perform about an hour of maze creation work per day... Every day. That includes days when I'm sad... Days when I'm angry... Days when I'm stressed out... That amount of work is plenty for me and that doesn't include any work involving content presentation; Any work involving this website is counted as 'content presentation.'

Part of the reason why I don't perform hours upon hours of maze creation per day is because, with more work comes the greater possibility of mistakes. Some mistakes are tiny; Others have been far larger. And none of those mistakes are fairly obvious; I currently don't have a lot of methods to double-check my work to see if I've made any errors along the way. Even if you fall off of a very tall cliff, gravity will eventually force you to collide with the ground below. With unicursal maze creation, if you make a mistake, that mistake could linger for years before being discovered... If it's discovered at all. Instead of falling off of a cliff, making a mistake here is the equivalent of floating in space: There's no gravity to assist you in seeing the error of your ways.

Today, I found a tiny mistake while performing analysis of the mazes. Six mazes were mislabeled in the 0505-13 determined field. It wasn't exactly a huge crisis and it was quickly resolved.

Yet the next discovered mistake might be a huge one. I just don't know how extensive the next mistake might be. I have few tools to discover mistakes except for going back and examining them in various ways. Examining prior mazes takes time... Time that, legitimately, could be spent on creating mazes. However, it's important to be accurate and I'd rather be accurate than fast.

There's a reason why I haven't completed the 5x6 determined field yet; Slow and accurate is always better than fast and sloppy. I'm heartened that I haven't seen any errors since I've slowed down my pace but that doesn't mean that I won't find one eventually. I hope that this is the last error out there... But I know that hope needs to be backed up by constant examination and research.

2023-03-07: The first ever exhibit is now on display in the Gallery. While I won't make an announcement here for every exhibit, I thought that it would be noteworthy to mention. Hopefully, this exhibit will be the first for many to come. On current display are examples of different types of Unicursal Mazes for you to enjoy.

2023-03-05: Every path that I create is special. It's like that phrase, "As a parent, I love all of my children equally." Simply because I dwell on one path or a series of paths more than others doesn't discount those other paths in any way.

Just today, while analyzing path 0506-05-33212 (formally titled, "0506-00g-33212," a determined path) in order to catalog it, I stopped for a moment to appreciate it. There's nothing overly unique about that path; It doesn't contain an unusual amount of corners or decision points... It doesn't belong to an original pair that's significant in any unique sort of way.

I was hit with the thought that it has taken me years to be able to draw that particular path. YEARS. Certainly, if I had worked faster, I could have gotten to that path sooner. Yet, speed isn't the point; The point is that, for all that I know, I may have been the first one to have ever drawn it. Certainly, I'm the first one to have drawn all (or, at the very least, most) of it's predecessors.

It's taken a lot to get there... And to where I am now. The 5x6 field has lasted years for me and I'm beginning to reach 'the end.' Granted, I'm not breaking any speed records but the goal is accuracy, not speed. And the goal is self-education, not false records of achievement. I've earned that path through drawing all of the other paths before it... Learning all of the techniques and procedures to draw that path by drawing all of the predecessors. I didn't push some button on a computer and let some computer program draw them out.

I can't predict when I'll finish the 5x6 field (maybe this year but... Who knows?) but, when I do, I'll certainly be a different person than when I started. I appreciate all of the paths that I've drawn.

2023-03-04: I don't have a content creation problem, I have a content presentation problem. I create Unicursal Maze content for about an hour (and, nowadays, longer) every day.

It's been very difficult for me to mentally justify the time to spend taking that content and re-packaging it into a format that's presentable and understandable for someone other than me. First, the mazes themselves are a bit miniscule when it comes to pixel length and width. I've learned that I've had to "upscale" those pictures, both to make them more viewable and more presentable.

It's never easy to admit fault in anything. My interests have a limited audience and my skills at presenting that interest are limited still. But I am learning to adapt, albeit slowly. After all, the website was re-designed...

2023-03-03: I'm always slightly paranoid whenever I encounter original pairs such as #56 (05F) where they don't yield a lot of completed paths. You always wonder whether or not you've missed a decision point somewhere. I haven't discovered a lot of mechanisms for error correction yet. It's a sore topic in my research; If I make a mistake, I may never know about it for years (or even ever) and this is why I go as slow and cautious as I do nowadays. Yes, I've seen some patterns in the data but those patterns don't tell me everything. It's a 'Which came first? The Chicken or the Egg?' scenario - You need the data to make the tools for error correction but, in order to make the data, you need to make few to no errors.

2023-03-02: I don't think original pair #56 (05F) for the 5x6 field is going to yield a lot of completed paths. I've spent two days working on it (14 cycles total) and I'm already yielding completed paths. The sooner that you start getting completed paths, the less completed paths overall from that original pair that you get. I think that I'm going to get less completed paths from #56 than I did from #55. Is that a guarantee? No, but it just feels that way at the moment.

2023-02-25: Welcome to the re-opening of Unicursal Maze Research. I hope that you enjoy your stay.

For returning patrons, you shall obviously note that the decor has changed considerably. It takes great humility in order to admit that my web design skills are quite lacking for the 2020s. The prior web design may have passed muster in the 1990s but, for today's standards, an update was in order. On this occasion, I have borrowed far more than I have invented; The world of web design has since far surpassed my ability to keep up and it's time for me to learn all of the new tricks while borrowing from those who already know them. Your patience is greatly appreciated in the meanwhile.

There are many changes as one may see, too numerous to elaborate on in one post. "The Path Forward," the website's 'blog,' is still here. However, it shall concentrate on what it was always meant to do: Be my way of expressing aspects of Unicursal Maze Research. For those interested in when I achieve completion of some aspect of that research, there is a new page called exactly that: "Achievements."

The Gallery is also as described, being a section where one may view paths of every type, for amusement, education, enjoyment or any other interest.

The Learning Center is where one may learn several aspects of Unicursal Maze Research. Prior patrons will see that the 'Primer' is there with very little change except for, of course, the updated visuals.

Finally, About provides information 'about' the website and other semi-biographical aspects of the website itself. It will also have contact information and a section detailing the occasional technical changes made to the website.

As with all aspects of the Internet, everything on this website is subject to change. However, I think that this current line-up will persist for some time. Again, I thank you for visiting and I hope that you enjoy your time here.