Proof I came up with some weeks ago. If a ruler covers the entire line and has marks at the dyadic rationals (multiples of a power of 2) that are larger if the power of 2 involved is higher, then any open interval has exactly one mark bigger than all of the others.
So for those of you who didn’t see the big link above this text, I got a channel on twitch.tv! I’ll mostly be using it to stream The Binding of Isaac for now, but I’ll most likely be playing some other games on there as well, so check it out!
I just proved a couple of interesting theorems relating to a concept I’ve defined and call the sequential discontinuity. Anyone who has studied calculus knows of three or four types of discontinuity. The first is the removable discontinuity, where a function is undefined or “wrongly” defined at a single point. The second is the jump discontinuity, where a function approaches a different value from either side of the discontinuity. The third type is the essential discontinuity, which is basically anything else, so teachers tend to further subdivide it into vertical asymptotes (which make up the vast majority of essential discontinuities) and oscillating discontinuities such as the discontinuity of sin(1/x) at x = 0 (even if the function is altered such that f(0) = 0). However, not all essential discontinuities fall into this category.
Consider the following function: f(x) = 0 for all x such that log(x) is not an integer and is undefined otherwise. (This is the base 10 logarithm). This function has obvious removable discontinuities at x = 10^k for integer k, but it also has a discontinuity at x = 0. This is because the formal definition of the limit requires an open interval of x-values around 0 where all f(x) except possibly f(0) are undefined, and no such interval exists. In layman’s terms, you can’t put a mark anywhere past 0 on the number line without a power of ten between 0 and the mark, no matter how close it is to 0. This discontinuity is not an asymptote and does not involve oscillation (the value of f(x) never changes where it exists).
A sequential discontinuity exists in two cases: at the limit of a convergent sequence of x-values where the function is undefined, and at the limit of a convergent sequence of x-values where the sequence of the corresponding function values is unbounded. Note that the second case especially can be a different kind of discontinuity as well; for instance, the discontinuity at x = 0 for f(x) = 1/x is sequential, as we can define the sequence x = 1, 1/2, 1/3, …, which converges to 0 and has function values that increase without bound. It is also a vertical asymptote, a simpler case.
The interesting thing about sequential discontinuities is that sometimes they are “removable” in the sense that the function containing them could be made continuous by changing countably many values (strangely not including the value at the discontinuity). However, they have undefined limits and as such are not removable.
I have here a link to a PDF proof and more formal definition. One small lemma is required for the second case but is not proven, so I will prove it here. An unbounded sequence has for every real number k and natural number M a value m > M such that a_m > k. To prove this, choose i such that a_i > k. If a_i > M, then m = i. If a_i <= M, then let K be the maximum of a_1, a_2, …, a_M. We are guaranteed K >= a_i > k. Then there must be a j such that a_j > K, and j must be greater than M (otherwise K would not be the maximum), so let m = j and a_m > k.
So I decided yesterday to get my own website! I’ll post the finished versions of all of my games there, as well as more info about myself, etc. Check it out!
Yes, another game release within the span of a few weeks. As it turns out, this is a game I started in 2011. It was up to release candidate 3, but then I lost access to Windows computers in general. No worries; I’ve been able to work out the last few bugs and get the game out.
The game is a card game based on the classical/Greek elements (earth, air, fire, and water) and otherwise plays similarly to Mau Mau or Uno. Single-player and multiplayer modes are included (both local and online). There is also AI. Overall, it’s a pretty fun game, and it has what I consider some of my best music in it as well.
The game is a standalone .exe in a .zip archive; make sure to unzip it first in order to get the DLL files out. The soundtrack .mp3s are included if you want it in iTunes. Let me know if anything goes wrong.
What you see above is both the logo of my newest game, called Resonance, and every possible screenshot from within the game. If it doesn’t look like it’s loaded yet, it probably has. The game takes place in complete darkness.
Resonance is a game about solving puzzles without the luxury of sight. The game takes place in a maze on a standard tile grid (tiles are around 4’ wide in my mind). Instead of being able to see the maze around you, you can hear what’s in front of you, represented as pitches relative to a base of 440 Hz. Distance corresponds to the pitch and volume of each sound, and object type corresponds to the timbre.
The game has 50 levels ranging from pretty difficult to infuriatingly difficult. Headphones and graph paper are both recommended.
In theory, a blind person could play this game without being hindered (as long as he or she could find the control keys). A deaf person, on the other hand, would know absolutely nothing about the state of the game.
I’m now on YouTube! Find me at http://www.youtube.com/user/Whirligig231. I’m not entirely sure exactly what will go up on here, but it’ll be worth a look.
So, here I am again, after another period of inactivity, and guess what! It’s time for another game review! This time, I’ll be reviewing Terry Cavanagh’s Super Hexagon. Check it out if you haven’t done so already.
First and foremost, pricing. The game is currently on sale for $0.99, but it won’t be for very long; Terry’s jacking up the price to $2.99 after a couple of days, so get it now. Not like it matters. I would pay at least $5 for this game if necessary. Yep, it’s worth more than a footlong from Subway. Price shouldn’t be an issue for anyone except maybe the “OMG IT HAS TO BE FREE” people (like me). So why is it more than worth the price? Well, let’s see.
Before I get to the actual game, you should know a bit about where I’m coming from with this review. I’ve known (in the Internet sense) Terry for quite some time by now; I helped to contribute one of the custom levels that ships with VVVVVV 2.0. I’m also currently as far as I know the world record holder for the highest score in the original Hexagon; let me know if you can beat 870:59.
Yes, the original Hexagon; Super Hexagon is in a sense a “sequel,” but that’s like saying Windows 7 is a sequel to Interface Manager. The game is essentially what Hexagon would have been if Terry had put enough effort into it to make it a full-fledged game and not just something created on the spot and left alone. Super Hexagon is the game that Hexagon wants to be when it grows up.
I’ll briefly describe the gameplay for those who haven’t played the original Hexagon (which is still free to play here). Basically, you play as a tiny arrowhead trying to dodge advancing walls of doom that fly past you into the hexagonal pit of doom at the center. You use the left and right arrow keys to rotate counter-clockwise and clockwise, respectively. Sounds simple, right? Except the entire thing is rotating. And flashing through a slew of warm colors. And pulsing in and out to the beat of the music. Yeah, it’s pretty trippy.
After crashing into the one-hit-kill walls, you get graded based on how long you survived. The ranks follow the progression Point, Line, Triangle, Square, Pentagon, Hexagon, which annoys the math section of my brain ever so slightly because it changes from simplices of increasing dimension to polygons of increasing side count. And Line ought to be called Line Segment. I mean, yeah, Terry could have changed this, but I doubt anyone would know what Henagon is supposed to mean anyway. And to be frank, I doubt anyone else cares. Once you’re past Hexagon (one minute of continuous survival), it’s all for the bragging rights.
Super Hexagon is an expansion of that same concept. You start up the game, but this time, you are greeted with a menu. You now have three difficulty levels to choose (later six), which start on Hard and end on Hardest. The additional levels get more absurd (level 5 is “Hardestest”), and each level has a personality and name of its own.
First is Hexagon. This is just like the original game in a lot of ways. It looks similar, the music is still Courtesy by Chipzel, and it plays mostly how it did before. However, there are some changes to notice. The walls in Super Hexagon are no longer one-hit-kill walls in all cases; it’s implied that what actually kills you is the pit. This means that you can slide along the sides of walls without breaking a sweat. I mean, without dying; you’re likely to break a sweat anyway when you hit one of the wall combinations that requires this move.
There are subtle graphical differences. The flashing colors go a bit lighter, and the game adds perspective distortion. But the main difference is a bit of a surprise that hits when you’re aiming for one of two gaps and it closes in front of you. Seriously. You’re now in a pentagon, not a hexagon. Sometimes, it’ll even become a square. This is a bit tricky (especially since a wedge is 3/2 the angle now), but it adds a new level of fun.
There’s another difference. Levels come in pairs. Once you hit 60 seconds, not only have you obtained the Hexagon rank, but the level changes. Now, the warm colors are cool. And everything is moving about three times as fast. You have entered Hyper Mode. Hyper Mode for Hexagon also becomes unlockable as the fourth level.
The other levels are just as fun, though a bit less original. Hexagoner has neon colors on a dark background and features a slightly more insane track from Chipzel. The pentagon and square parts are gone, so all you have to worry about is dodging the spirals of walls. Now, normally the wall slide move makes spirals six times as easy (see what I did there?) in Super Hexagon, but new spirals have been inserted to take their place as the most infuriating wall combination ever.
That is, until you reach Hexagonest. Then, walls will come in a barrage from all directions. The music is faster, and everything flashes through the entire spectrum. The entire level will lurch through a half-turn in a split second, disorienting you more than a teacup ride on the International Space Station. With all the rainbow colors and the potential to kill a small child with its difficulty, I’d say this is the video game level equivalent of John Wayne Gacy. But that’s a title probably reserved for Hyper Hexagonest, a level whose horrors I haven’t even glimpsed to this day.
This game is very addictive, and I give Terry a lot of credit, but there are a couple of small nitpicks I have to cover. One, what happened to the humming noise in the menu screen? Hexagon had this mechanical hum of sorts while you cool down between runs, but Super Hexagon has only silence. Two, I’d like to see some sort of analysis screen. Something about how many times you’ve died and perhaps some score calculations would be nice. Finally, a bit more documentation on how the game works could help, but this is really getting into details. The game is pretty much a “pick up and play” game, although with the way Terry likes his difficulty curves, it’s more like “pick up and get a hexagon shoved up your butt.” I don’t mind the difficulty, though I’m sure a lot of people do (the reviews on the App Store seem to indicate as much). I like the triumph of having progressed in skill from a game where 15 seconds is an achievement to a game where 60 seconds isn’t anything to be proud of.
So, how about some number ratings? I’ll use a 0-6 scale for these, and give a brief explanation as to why.
Gameplay: 6/6. This is one of the most addictive games you’ll play, and it’s a great way to waste time. Unless you don’t like the difficulty, you’ll find yourself with no reason to ever put it down.
Controls: 6/6. Really, you can’t go wrong when you only need two movement directions and nothing else. Sure, it may be slightly confusing when you’re at the bottom and they seem reversed, but you get used to that soon enough.
Interface: 6/6. My only complaint would be that there’s no button to exit a game in progress, but you can just let the game sit for 2 seconds and automatically accomplish that by losing.
Levels: 4/6. Yes, there are three (or six) levels. No, I don’t see too much variety in them. It would help if Hexagoner and Hexagonest had their own unique gameplay elements like Hexagon’s shape changing, but they don’t. It’s all the same wall-dodging. Apart from that, though, the difficulties are a mostly logical progression (you can argue that Hyper Hexagon is easier than Hexagonest, but they’re separate groups of levels anyway).
Graphics: 6/6. Terry’s done all that he could with this simplistic style and done it well. If you have epilepsy or motion sickness, this game is not for you.
Audio: 6/6. Apart from the minor “ambient noise in the menu” issue, there’s nothing wrong with the audio in this game. Terry knows how to pick his chiptune artists. Now he has me wanting to hear what a collab between Chipzel and SoulEye would sound like.
So total: 34/36 = 5.7/6. If you’re one of those mainstream raters, this is equivalent to 4.7/5, 9.4/10, 94.4/100, etc. Overall, a great game, one of the best iPhone masterpieces I’ve found. (Although that might not be saying much.) Terry, you’ve done it again. Although given your penchant for odd names (e.g. VVVVVV, ____), I would’ve expected you to call it ⬡. Or 0x2B21. Something like that.