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Silly Druid

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  1. Silly Druid

    Why Tau is better than Pi

    By Silly Druid,


    Mathematics
    I'm writing this on Pi Day, but I think pi makes no sense as a fundamental mathematical constant, and it's used only because of force of habit.
    The circle is defined as the set of points on the plane with a given distance to the center. That distance is the radius. So the "circle constant" should be defined using the radius, not diameter. It has been proposed that this constant would be denoted by the Greek letter tau. Tau is equal to 2*pi.
    The full angle (360 degrees) in radians is one tau instead of two pi, which is a more simple and natural way of doing things.
    Also Euler's identity is in my opinion even more beautiful when using tau instead of pi:
    e^(i*tau) = 1
  2. Silly Druid
    I've read about this in an old book about Bridge (the card game). The example below is a modified version of the one used in that book.
    There are 3 people (including you) sitting by the table, and every one of them gets a card. There are 2 black cards and one red card, and for some reason you want to know who has the red card. The cards are lying on the table, face down. So, how much information do you need to know who has the red card?
    By definition, the amount of information (in bits) is log2 of the number of possibilities. For example, a byte has 256 possible values, which means it contains log2 (256) = 8 bits of information. In this case, you have 3 possibilities (you know one of the 3 people has the red card), which means you need log2 (3) ≈ 1.585 bits of information.
    Now, you reveal your card and see that it's black. How much information did you gain? Well, about the card itself you gained one bit, because it could be black or red. But about the question "who has the red card" you had 3 possibilities and now you have 2 (one of the two other people has the red card), which means now you need log2 (2) = 1 bit of information. So the information gained is 1.585 - 1 = 0.585 bits. So getting a smaller amount of information than one bit is indeed possible!
  3. Silly Druid

    -1/12

    By Silly Druid,


    Mathematics
    Some time ago there was some controversy across the Internet whether 1+2+3+4+... = -1/12 or not. Here's what I think about this:
    Clearly, this sum diverges to infinity, so you can't assign any finite value to it, using the standard math rules. But there is a way to associate this value with this sum, which uses some advanced concepts that I don't fully understand, like analytic continuation, Riemann zeta function, and Ramanujan summation.
    So I did something different. First, I took the formula for partial sums: S(n) = n * (n -1) / 2, and treated is as a function of all real numbers: f(x) = x * (x -1) / 2. It's a quadratic function, with a graph that looks like this:

    Between -1 and 0 the value is negative. I wanted to know the area between the x axis and function in that interval, so I took the integral. And guess what? The result is -1/12. Coincidence? I think not.
    Later I watched this video, and it confirmed my suspicions that something interesting is going on here. It's in the last minute of the video.
  4. Silly Druid
    Here is my (probably pretty bad) attempt at translating the lyrics to English:
    When I weaved my braids into winter days
    Blue ribbons smiled at you like two sisters
    And the gloves, which I kept losing so you would look for them
    Always had for you ten nuts hidden in their fingers
    And into the footprints on the road
    Which I made gliding through the snow
    I threw summer's fruits
    For you to follow me
    Today I don't have braids anymore
    The wind took the ribbons
    I don't wear gloves
    I don't gather nuts at all
    And there are no footprints
    Because it rarely snows here
    I save the fruits for worse days
    And you're still following me
    How can you just love me like that?
  5. Silly Druid

    On the existence of chairs

    By Silly Druid,


    Philosophy
    What inspired me to write this entry is this video. Before watching it I haven't realized there is so much controversy over a seemingly simple statement that objects like chairs exist. So let me explain what I think about all this.
    First, we need to define what "exist" means. What I mean here is physical existence, as opposed to mathematical existence, which is a different thing. My definition is as follows:
    If something exists physically, it means it's a part of a valid description of reality.
    But what's reality? According to Philip K. Dick, "Reality is that which, when you stop believing in it, doesn't go away." I think it's a pretty good definition, because it captures an important aspect of it: consciousness (there's more about it in my earlier blog entries). Without consciousness there is no reality. Here's my version of the definition:
    Reality is everything that can directly or indirectly interact with consciousness.
    Which means consciousness is also a part of reality, because it can interact with itself or with another consciousness. But wait, aren't there fictional concepts that can affect our consciousness, like these colorful ponies from MLP? Not exactly. They don't exist, so they can't affect us in any way. What can affect us are the depictions of these fictional concepts, which are existing physical objects. For example an episode of MLP that we watch on our screen is something that exists, while the MLP ponies themselves are not.
    Which leaves us with another question, what is "a valid description of reality"? Well, there are many ways we can describe reality. The most basic way (but not very useful in practice), is to take into account the most fundamental elements of it. According to our current knowledge, these are the elementary particles, like quarks, electrons and photons. But we can also think about larger structures built from these fundamental elements, like nucleons, atoms, molecules and so on. When we're not far away from the fundamental level, things (for example specific kinds of molecules) are pretty well defined. But when we move to larger structures like chairs, we need to use fuzzy logic, because there are objects that are "kind of like a chair, but not quite". This way we avoid problems described in the video, like the necessity of defining a boundary line between "chair" and "non-chair", which would mean it's possible to take away one atom from a "chair" to make it a "non-chair". When using fuzzy logic, these problems don't exist, because everything can have a degree of "chairness", and taking away one atom just changes it by a very small amount.
    So what we do is to create a more or less vague conditions of what constitutes as a specific kind of object, like a chair, and check if the reality meets these conditions or not (or meets them to some degree). If we observe that it meets them, then we can say that there's a chair. We can also create more "exotic" kinds of objects like "trogs" and "incars" mentioned in the video, the only difference is that they are less useful in practice than the ones we use, like chairs.
    So the conclusion is, I think chairs exist, as one of many ways to describe reality.
  6. Silly Druid

    How to Solve the Time Loop Paradox

    By Silly Druid,


    Physics
    In principle, physics doesn't forbid traveling back in time. We don't have the technology to do it, but maybe someday we will. But what about the paradoxes it causes? Well, if you go back in time and, for example, land in a distant galaxy where you don't do anything that would contradict what you already know about that place, then there is no problem. The paradox arises when you create a time loop, which means a closed chain of cause and effect.
    Now, if the time loop is consistent, then there is no paradox. Good examples of consistent time loops in fiction are the ones that happen in "Harry Potter and the Prisoner of Azkaban", and in the MLP episode "It's About Time". If every link in the chain fits the previous one, then it's all good. Although there is another apparent paradox, because this kind of loop means that some things can seemingly appear out of nothing. In "It's About Time" it's the information where the time spells are kept. Future Twilight tells it to Past Twilight, but how does Future Twilight know it? Well, when he was Past Twilight, Future Twilight told her. So there is no clear source of this information. But it's not actually a paradox, it doesn't break any laws of logic or physics, so we just have to accept that such things can happen.
    Actually, this kind of time loops happen all the time. There is a phenomenon called "pair creation". It's not only a theoretical concept, it causes an effect that has been measured. Even in vacuum, due to quantum fluctuations, a pair of "virtual particles" can appear. It's a particle and an anti-particle, they appear close to each other, and annihilate after some time. That's the standard description of the process, but, according to Feynman, an anti-particle is a particle moving back through time. If we take that into account, we can describe it as one particle moving on a closed loop through space and time. So, in the micro scale, the consistent time loop happens all the time. And these are the kind of loops where things (namely particles) appear "out of nothing".
    But what will happen if we have the ability to create a macroscopic time loop, and we deliberately try to make it inconsistent? Examples in literature usually include drastic measures, such as killing one of our ancestors. There are many theories what would happen in such case, and I have one as well. Note that I'm not a professional physicist, and it's based on intuition rather than any true knowledge, so it can be complete rubbish. But I think it's interesting enough to share with you here.
    First, let's consider the "orbitals" of electrons in atoms. Only some sizes and shapes of them are allowed, because the quantum mechanics requires that the wave function after "going around" the atom matches the one from the previous "round". Otherwise the wave cancels itself, which sets the probability of such process to zero. I think it's the same in the pair creation process, the virtual particle's wave function after going around the loop in space-time must match itself. And my theory is that it also happens in potential macroscopic time loops. If we try to set up an experiment that allows an inconsistent time loop, then we will observe some large scale quantum effects that will work like a force that prevents us from doing it. It's like the Pauli exclusion principle, where if we try to do something that is improbable in quantum mechanics, like putting two electrons too close to each other, we will get something that works like a repulsive force that prevents it. Actually I suppose that large scale time loops may be very hard to do in practice, because we would have to make sure that every particle involved in the process behaves in a consistent way. So I'm afraid these fictional examples I mentioned will remain fiction for a long time...
    Whoa, that was a long entry! But I hope you liked it, next week we'll talk about extra dimensions!
  7. Silly Druid

    More on Superdeterminism

    By Silly Druid,


    Physics
    The goal of this entry is to explain my thoughts about superdeterminism in a more detailed way.
    We should start with the double slit experiment: there is a source of particles, two slits, and a screen. The particles can go through either of the slits, and even if we emit a single particle at a time, after many particles there is an interference pattern at the screen, showing that the particle seems to go through both slits at the same time. But if we make a measurement to determine which slit the particle used, it always uses only one of them and there is no interference pattern. So what actually happens here?
    The most obvious explanation would be a simple hidden variables theory, stating that the particle already carries all information needed to determine the results of all possible measurements. So if we don't measure it, it goes through both slits, but if we measure it, it already "knows" which slit it should use. But this explanation is wrong. The Bell Inequality experiments prove it. It's simply not possible.
    Another explanation is the standard quantum mechanics (Copenhagen interpretation). As far as I understand it, it states that if we don't measure it, it's truly undefined which slit the particle (or rather its wave function) uses. It moves through both slits with the probability of 50% each, and it interferes with itself. But when we try to determine which slit was used, the wave function "collapses", so it becomes 0% in one slit and 100% in the other. The problem with this interpretation is that the "collapse" doesn't make much sense as a physical process, because it happens at the same time in all the space occupied by the wave function: if we detect the particle in one slit, the part of the wave function in the other slit instantly becomes 0. This is Einstein's "spooky action at a distance". And we don't even know what "instantly" means, because in special relativity the concept of "at the same time" depends on the frame of reference we use.
    So how does superdeterminism fix these problems? By assuming that the following two things are correlated: the path of the particle (let's call it A), and the way we measure it (B).
    Before watching Sabine Hossenfelder's video, I didn't believe in superdeterminism. Not because the free will nonsense, but because in the sources I've seen it was presented as if the correlation meant there is a common cause (C), that causes both A and B. This makes no sense, because there is no way a physical process would determine both the path of the particle, and our decision how to measure it.
    But now I realized there are other options as well. A correlation can also mean that A causes B, or B causes A.
    "A causes B" makes no sense either, because again there is no way a physical process would make the path of the particle cause our decision how to measure it.
    But "B causes A" is an interesting possibility. If the way we measure a particle affects its path, then we can explain the "problem of measurement" without the need of "wave function collapse". Instead of collapsing, it's already created in a way that depends on what we measure (or don't measure). So if we don't detect which slit the particle uses, the wave function goes through both slits and interferes with itself, and if we do detect it, it goes through one slit. There is no collapse, and it's also consistent with the Bell Inequality results.
    But the measurement is made after we emitted the particle and it has "chosen" its path. Which means backwards causality: the effect (particle choosing its path) happens before the cause (the measurement). The way I see it is when we try to do something with a particle (which means trying to emit it, absorb it, measure it etc.), it creates a "vertex" on the grid of all possible particle interactions, and these vertices have a probability to be connected with each other by "particle paths". I think it's something similar to the "transactional interpretation" of quantum mechanics.
    [EDIT] An important thing is why this backwards causality doesn't cause time paradoxes. It's because it affects only the wave function, which is immeasurable. We can't actually see what happens with the wave function before we measure it. So you can argue that it simply doesn't exist at all. But it should exist, because it's what the Schrödinger equation describes, in other words it's what makes the entire quantum mechanics work. Without it (or some equivalent of it) we don't understand what happens at all. So my understanding of superdeterminism implies backwards causality, but in a way that doesn't allow you to actually send any information back in time.
    As for the tachyons, for those who think they are forbidden in special relativity, it's not true. If we assume a particle has an imaginary mass, it must move faster than light. But now I'm not so sure they are what really happens here, maybe it's something different, like wormholes. (Yes, there is a conjecture, called "ER = EPR", which states that quantum entanglement creates a wormhole between the particles.) What I mean here is that if we successfully combine two theories, it usually predicts something new. For example, when Dirac combined quantum mechanics and special relativity, it predicted the existence of anti-matter, which was experimentally confirmed later. So if we combine superdeterministic quantum mechanics with general relativity, who knows what we will find?
    (Sorry if this entry is too long and messy for you, you can ask questions in the comments so hopefully I can explain some things better.)
  8. Silly Druid

    Superdeterminism

    By Silly Druid,


    Physics
    I just watched a video from Sabine Hossenfelder: https://www.youtube.com/watch?v=ytyjgIyegDI
    She talks about an idea that can fix some problems with our understanding of quantum mechanics. And I have some things to say about it. First, I'm glad I'm not the only one who doesn't believe in all that free will nonsense. Second, I think the idea of superdeterminism has some very interesting implications.
    It states that the path of the particle depends on what you measure. But what she doesn't say in the video is that the measurement takes place after the particle "chooses" its path. Which means we have causality that works backwards in time.
    She also mentions that superdeterminism can be combined with general relativity. So, if we do it, I think it will imply the existence of tachyons, particles that always move faster than light, and can carry information back in time. But their function should be somewhat limited, to avoid time paradoxes. Maybe this will be a part of the long awaited theory of quantum gravity?
  9. Silly Druid

    Adventure Awaits!

    By Silly Druid,


    MLP Fan Works
    Today I'd like to recommend a choose-your-own-adventure series that I really like: d20 Pony. It's a long running series, in an expanded universe based on MLP G4, but with new areas to explore and species to meet. It's a really fun and exciting story with epic adventures and very likable characters (especially the main two: Trailblazer and Moonflower). It's mostly based on OCs, but there are also some canon character appearances.
    It works like this: anyone can post a command saying what the protagonists should do next, then one of the commands is chosen at random and executed, and the next part of the story is revealed, together with a 16-bit style picture. Sometimes there are dice rolls, used to determine if an action succeeds or not (hence the name). As you can see with the current entry, I'm taking part in it too (I'm "nopony" there). It was a part of my inspiration for the "Champions of Celestia" roleplay (the other part are my Tails of Equestria sessions, hosted by @abrony-mouse).
    Even if you're not going to take part in it, I highly recommend that you read it from the beginning, and treat it as an illustrated fanfic. (Warning: Sometimes it gets a bit adult-themed. It's nothing explicit though, just things like a vague description of the situation.)
  10. Silly Druid

    The Dividing Point

    By Silly Druid,


    Mathematics
    It all started with a seemingly simple question I saw somewhere: Is it possible to divide a square into two triangles? You might think you just need to cut it through the diagonal and you're done. But, if you divide something, its every point must belong to one part or the other. So what about the points on the diagonal? To which triangle do they belong? This problem seems to me like something artificial and counterintuitive, and makes me think the rules of geometry should be modified. Normally a point can't be divided into parts, but let's change the rules and assume it can.
    Let's start with a simple 1D object: an interval on the number line. Its ends can be either open or closed. Closed means that the ending point belongs to the interval, and open that it doesn't. But if we can divide a point into two parts, then we can have an interval where an ending point is "at the edge", and only half of it belongs to the interval. It's like taking limits, where we can approach a value from one side or the other, and in some cases we get different results.
    In case of 2 or more dimensions the situation is a bit different, because instead of two parts we can have any number of them. It's like a pie chart - when we zoom in to its central point, the situation around it doesn't change: the area is still divided into several parts, with the same proportions as the entire pie chart. So we only need to apply this division to the central point itself. In case of dividing the square into two triangles, the points at the diagonal belong in half to one triangle, and in half to the other. The two points at the end of the diagonal (in the corners of the square) only belong in 25% to the square, so after making the division they belong in 12.5% to one triangle, and in 12.5% to the other.
    What do you think about my version of geometry, does it make any sense to you? (It's my original idea, if you know something like it that already exists, then let me know in the comments.)
  11. Silly Druid
    (This entry contains information about the "Champions of Celestia" roleplay. If you're not interested in this RP, just ignore it.)
     
    1. Attributes
    You have 4 attributes:
    Strength - your physical strength and constitution. Used for moving and carrying heavy objects, bucking, resisting some physical conditions etc. In combat used to damage the enemy.
    Agility - your reflexes, and the speed and coordination of your moves. Used in situations where a fast reaction or skillful motion is required. In combat used to avoid enemy attacks.
    Intellect - your knowledge and reasoning ability. Used in intellectually challenging situations, and for some spells.
    Charm - the ability to influence others, and to resist being influenced. Also used for mind-affecting spells.
    Starting values (Strength-Agility-Intellect-Charm):
    Earth Pony 4-3-3-4 Pegasus 3-4-3-4 Unicorn 3-3-4-4 You have 2 points to increase the attributes when creating a character, and 1 point per level up. The maximum value of an attribute is 10.
     
    2. Energy
    These two stats have a maximum and current value. Between adventures you go back to the max values. They can be also recovered during an adventure, for example by resting.
    Stamina - your physical energy. Max value is Strength+Agility. Can be lost in combat or due to fatigue or other harmful physical effects. If it reaches 0, you are unconscious.
    Willpower - your mental energy. Max value is Intellect+Charm. Can be lost due to mental influence of hostile beings. If it reaches 0, you are "under their spell".
     
    3. Skills
    You start with 3 skills, and you can have up to 6 of them. The first skill depends on your race (but if you really want to create a character that doesn't have it, for example a pegasus that can't fly, like Scootaloo, then it can be replaced with another skill). The second skill is your Cutie Mark talent. The third one, and the ones you learn later, are "extra skills". The extra skills can't have higher values than the cutie mark skill.
    Racial skills:
    Earth Pony - Inner Strength (the ability to quickly recover from negative conditions and other bad situations)
    Pegasus - Flight
    Unicorn - Telekinesis (the ability to move things at a distance)
    Some examples you can use as cutie mark talents or extra skills (taken from the Tails of Equestria manual):
    Creative skills: Baking, Comedy, Drama, Fashion, Music, Painting, Sculpting etc.
    Knowledge: Art, Biology, Chemistry, Geography, History, Law, Magic, Math, Physics etc.
    Physical skills: Bucking, Climbing, Dancing, Juggling, Jumping, Running, Sneaking etc.
    Unicorn spells: Forcefield, Locate, Stun Ray, Teleport etc.
    Cloud Wrangling (pegasi only)
    Pony Sense (like Pinkie sense)
    Healing
    Speak with Animals
    The Stare
    You're not limited to these skills, you can invent your own ones. It can be anything you want, within reason of course. Also keep in mind that it's medieval times, so skills related to advanced technology won't be useful here.
    The starting value for racial and cutie mark skills is 4, and for the extra skills it's 3. On level up, you have 2 points that you can use to increase existing skills or (if you have fewer than 6) learn new ones. Max value is 10, the same as with attributes.
     
    4. Tests
    Tests are how you use your attributes and skills in practice. They are performed by rolling a die with a number of sides equal to the value of the attribute or skill in question. Sometimes you will roll both an attribute and a skill, and use the better result of the two. Your degree of success depends on the result of your roll and the difficulty of the test. I will roll the dice for you (not physical dice, I'm using this page for it), when the situation calls for it, and post the results and their consequences.
     
    (This manual will be updated when needed.)
  12. Silly Druid

    Complex Numbers: The End of the Road

    By Silly Druid,


    Mathematics
    I'm going back to math as promised. Let's think about how and why the different kinds of numbers were created.
    First there are "natural numbers". It's a concept that is easy to understand, because it can be used to count physical objects we see around us. We can also make some simple operations on them. We can add them - no problems here, because if we add two natural numbers, the result is also a natural number. But we can also subtract them, and that's where a problem arises: if we subtract a number that is bigger than the one we subtract from, the result is NOT a natural number. So what can we do? We can invent a new kind of numbers, called negative numbers, that are the results of such operations.
    The same goes with rational numbers - they were created as the extension of the set of numbers in case division doesn't give us a whole number. Irrational numbers are used to solve some equations, as well as geometrical problems, like calculating the diagonal length in a square. There are also transcendental numbers, that appear in other kinds of operations, like calculating the length of a circle.
    So does this process ever end? Yes, it does. The end of it are the complex numbers. They allow all kinds of operations on all kinds of operands. We can take a square root of a negative number, a logarithm of a negative number, and so on. Almost everything is possible. (Things like dividing by zero are still impossible, but that's another story.)
    Some results of these operations are multi-valued. But is this something that only the complex numbers can do? For example, everyone (with basic math knowledge) knows that the square root of 4 is 2, because 2 squared is 4. But -2 squared is also 4, so it should be another value of the square root of 4. It's just a convention that we take only the positive value, and not the negative one. So operations with multi-valued results are not exclusive to complex numbers. Another thing is that some of these operations are quite complicated (look for the formula for a+bi to the power of c+di, it's total mess).
    There are further extensions to the set of numbers, like quaternions, but they are more like artificial constructs. They are not needed to make any operations possible, they are just created to serve some purpose (quaternions are used to represent rotations in 3D space, for example). So complex numbers are the end of the process of making the set of numbers complete. And they appear in many areas of physics, so it seems nature uses them a lot. So I think they are the ones that truly deserve to be called "natural" numbers, and using names like "real" and "imaginary" (for the two parts of a complex number) is just wrong, and is the result of superstitions from the time they were introduced.
  13. Silly Druid
    Before we continue our little list, some honorable mentions:
    Cotton Sky - I really like her mane.
    Daring Do - I don't know why, but I love her voice.
    Flurry Heart - most unused potential. I imagine her as a future ruler, with a conflicted personality.
    Lily Longsocks - stronk.
    Twilight Velvet - best mom.
    And now, the top 10:
    #10 Cutie Mark Crusaders. Good depiction of children, who are enthusiastic and eager to do things, sometimes they mess up, but their hearts (as strong as horses) are in the right places.
    #9 Zecora. A fine character is she, the mysterious zebra who lives in the Everfree.
    #8 Sunset Shimmer. I think it's the combination of appearance, personality and voice acting, that makes me feel that there is some kind of 'warmth' emanating from her, if you know what I mean.
    #7 The Great and Powerful Trixie. Undoubtedly the greatest pony who ever lived.
    #6 Starlight Glimmer. What I like about her is that her personality is not easily defined by one or two traits, like most of the other characters. Also I love her nervous laughs.
    #5 Maud Pie. Relatable, as I express my emotions in a similar way as her (mostly by not doing it at all). Also I love her sense of humor.
    #4 Twilight Sparkle. Highly relatable, as I'm a studious and socially awkward person myself.
    #3 Fluttershy. Another relatable one, because I'm a shy person. Also the embodiment of cuteness.
    #2 Celestia and Luna. I put them together to avoid taking sides in the debate, which one I like more. Beautiful and majestic princesses with interesting personalities, what's not to like about them?
    #1 Marble Pie. Cute, charming, mysterious. Best pony.
    PSA: If your favorite characters are not in the list, it doesn't mean I don't like them, they just didn't make it to the top 20. I like most of the characters in the show.
    Thanks for reading, next time we'll go back to the usual kind of content in this blog.
  14. Silly Druid
    As I mentioned earlier, I wanted to do something special for this time of the year. After a long consideration I decided to post the list of my favorite characters in all of FiM and EqG (I think it's too early to rate the G5 characters). Note that some entries are groups of characters instead of single ones, because I particularly like them as collective entities. So here we go:
    #20 Philomena. I really wanted to see more of her.
    #19 Plants and inanimate objects. Boulder, Phyllis, Smarty Pants, and so on. I think they add some great humor to the show.
    #18 Sugarcoat. Her brutal honesty impresses me.
    #17 Lyra & Bon Bon. I especially like Lyra after reading "Anthropology." Bon Bon as a secret agent is a very good idea too. And I think they complement each other perfectly.
    #16 Big Mac. Eeyup!
    #15 Cheerilee. Good teacher, charming personality. Underrated character.
    #14 The Student Six. I think they are a good addition to the show, I like their personalities and dynamics.
    #13 The "Legion of Doom". I feel in love with them after seeing "Frenemies". And I just had to make an ask thread for them...
    #12 Rarity. She is quite charming, and often unintentionally funny.
    #11 The Dazzlings. Best. Villains. Ever.
    I'll post the second part around the New Year. Any guesses who will be in the top 10?
  15. Silly Druid

    Not All Magic Was Gone

    By Silly Druid,


    MLP G5
    Before we go back to math, I decided to address something about the G5 MLP movie, and I think it's a good time to do so, because we were just talking about magic in my previous entries. So it would seem that all magic had disappeared from the world, before Sunny and her friends brought it back. But I think it's not true, and here are some examples of magical things that were still in effect:
    Cutie Marks - ponies still had them, and according to FiM lore, they are magical. The "sparkle" or "luminescence" that Izzy, and probably other unicorns, can see. It does sound like a magical effect. And I hope we will learn more about it in the upcoming series. Hitch's bond with animals, it looks like he has similar abilities as Fluttershy (also in some official materials his Element of Harmony is Kindness). When Izzy first meets Sunny and says "Hi, new friend! My name's Izzy!", there is a little spark of magic on her horn. I think a small amount of magic remained within the ponies, and when two ponies of different races had a friendly encounter, it reacted to it, showing that it was the way to restore all magic to the world. That's all for now, the next entry will be around Christmas, so I think I'll do something special for it, but for now I'm not sure what it should be...
  16. Silly Druid

    Explaining Magic, part 3

    By Silly Druid,


    Fictional Concepts
    Time for my physical theory of magic. As a reminder, it's totally fictional, applies to fictional universes like MLP, and I don't think that's how physics works in real life.
    For simplicity, let's imagine the universe as a two-dimensional object, it will be easier to think of its shape that way. So what shape can it be? It can be like a Möbius strip, which has only one side, but it can also be like a normal sheet of paper, which has two sides. (But it doesn't have edges, so you can imagine that when you go to the left, you emerge on the right, like in Pac-Man games. And when you go up, you emerge at the bottom.)
    So what if the other side is a universe of its own? Moreover, a universe different than ours. The laws of physics, and the kind of objects that can be found there are different, but most importantly, time flows in the opposite direction. By which I mean the thermodynamic arrow of time. So if there are any sentient beings in that "mirror universe", their past is our future, and vice versa. Then magic can be a force that can break the barrier between the two universes, and allow transferring some things between them. This will explain the ability to break the second law of thermodynamics and to go back in time. And some other magical effects can be caused by the differences between the laws of physics in the universes. I know it's not a highly developed theory, and it doesn't explain every magical effect in detail, but I think it's a good way to think about how magic works.
    Next week I'll probably go back to math, but I'm running out of ideas, so if you have anything in mind that I should write about (I mean things of similar kind to the ones I already did in this blog), then go ahead and propose it in the comments.
  17. Silly Druid

    Explaining Magic, part 2

    By Silly Druid,


    Fictional Concepts
    As previously established, I need something that makes magic special, that allows it to bypass a physical limitation that normally applies. And I think the best choice for that limitation is the second law of thermodynamics. It states that in a closed system (and the entire universe is a closed system) the amount of entropy (which can be understood as disorder) must remain the same or (more likely) increase. It also means that some processes are very easy in one direction of time, and very hard or impossible in the opposite direction.
    A good example is breaking things - it's easy to do, but making them whole again can be very difficult. And we have just this in the show - after Flurry Heart breaks the Crystal Heart, it is made whole again using magic. It can also explain some standard spells often seen in video games, like healing (mending the body is the reverse process of injuring it), or a fireball spell (heat tends to dissipate over time, so concentrating it in one place is the reverse process of it).
    There is also an unexpected implication of this interpretation of magic when we consider some things Discord does. In "Make New Friends But Keep Discord" he does something that looks like a time-reversed version of washing the dishes. Normally washing the dishes increases the overall entropy, so the reverse process should decrease it. Which leads us to the conclusion that Discord is actually the one that brings order to the world!
    Next week I'll propose a crazy physical theory that makes it all possible.
  18. Silly Druid

    Explaining Magic, part 1

    By Silly Druid,


    Fictional Concepts
    To make things clear: I don't believe magic exists in our world, I'm just considering it in context of a fictional universe, like MLP. I don't like the "It's magic, you don't have to explain it" approach. If magic is an integral part of a fictional world, then it should be explained like any other part of it. And that's what we will try to do here. We'll start with the sources of magic. What causes it? I think what Starlight says in "All Bottled Up" is a good explanation:
    Of course there are other things that need to be considered here, for example the unicorn horns seem to be a tool that helps focus the magic, and do things like forming it into a laser beam. But the basic idea that magic is fueled by emotions is the most important thing that we need to know. It explains why the magic of friendship and love seem to be the most powerful forms of it, as friendship and love are very strong emotions. Also, sometimes when we see ponies drained of magic in the show, they also seem to be devoid of emotions, at least to some degree.
    In the next installment we will talk about the effects of magic. What kind of physical limitations should it be able to overcome? If you have any ideas about that, let me know in the comments. I will share mine in the next entry.
  19. Silly Druid

    Real Numbers Continued

    By Silly Druid,


    Mathematics
    The title of this entry means we're still talking about the real numbers, but it also means we're talking about something called "continuum". But what is it? The answer is it's a kind of infinity, and it's different from the "countable infinity". (There are more kinds of infinity, but let's focus on these two.)
    "Countable infinity" is, for example, the number of elements in the set of all integers. "Countable" means we can arrange the elements in a sequence, so for any integer there is a well-defined previous one and next one.
    Rational numbers are an interesting case, because they can't be ordered in a sequence where every element is bigger than the previous one. Why? Because between any two different rational numbers we can find a third one, for example the arithmetic mean of them. But there are ways to order them in a different kind of sequence. We can put them in a big table, where the horizontal position is the numerator, and the vertical position is the denominator. Then we can go through the diagonals in this table, and add the unique fractions we find there to the list. Unique means we can omit the ones that are already there, for example if we already have 1/2, then we don't have to put 2/4 on the list. This way we can create a proper sequence, which means the set of rational numbers has "countable infinity" elements.
    In the set of real numbers, it's different. They are not a sequence, and there is no way we can make it a sequence. Which seems strange because in the previous entry I proposed an algorithm that generates them all, and at any finite step they form a nice sequence. But the exponential growth of the number of elements in every step causes something strange when we go to infinity: a sequence stops being a sequence and becomes a total mess. And that's continuum for you.
  20. Silly Druid

    Are Real Numbers Real?

    By Silly Druid,


    Mathematics
    Let's talk about the different sets of numbers. First, we have integers, which are a simple and intuitive concept. It's a series of numbers going in both directions (positive and negative) to infinity. Rational numbers are not hard to understand either, as the name suggests they are ratios of integers. But what about the real numbers? Are they a valid mathematical concept? There is one fact that made me question it: most of them are "unreachable", which means we cannot create any formulas that "pinpoint" them. The number of types of mathematical symbols is finite, and the number of symbols in a formula is finite too. This means that the number of all possible formulas is "countable infinity", while the number of all possible real numbers is "uncountable infinity". So most of the real numbers can't be expressed as formulas. This also applies to any continuous subset of them, for example the set of all real numbers between 0 and 1.
    But I still think the real numbers are a valid concept (or are "real" in Platonic sense). What convinces me is that there exists a way to construct them all. But instead of constructing one at the time (like we do with integers, where we're just adding one number after another to the set), to create an "uncountable infinity" we need to make the number of elements added in each step grow exponentially. For simplicity we're using the binary code, but it can be also done in decimal or any other base. The point I use here works like the decimal point.
    First we create 0.0 and 0.1. Then, in every step, we split every series of digits created so far into two, by adding 0 and 1 at the end. Here's how it works:
    step 1
    0.0
    0.1
    step 2
    0.00
    0.01
    0.10
    0.11
    step 3
    0.000
    0.001
    0.010
    0.011
    0.100
    0.101
    0.110
    0.111
    and so on.
    After an infinite number of steps, we will create all the real numbers between 0 and 1. Expanding it to all the real numbers is not a problem and can be done in many ways, for example by adding integers to them, or by changing the position of the point. This procedure convinces me that the real numbers make sense as a mathematical concept. Of course it requires an infinite number of steps (after finite steps we can't even create any irrational number), but it's understandable when we're dealing with infinite sets.
  21. Silly Druid

    The Guiding Light

    By Silly Druid,


    Culture
    Warning: contains spoilers for the G5 movie.
    This entry is different from what I usually post here, but I just had to share some of my thoughts about the new movie. Specifically, what I want to discuss here is the symbolism of the lighthouse. I love that they made it the house of Argyle and Sunny, and here's why:
    The thing that immediately came into my mind when I saw the movie is a piece of literature from my country: a short story called "Latarnik" ("The Lighthouse Keeper") by Henryk Sienkiewicz, where the titular character has a vast knowledge about the old times of glory of his country (in this case it was Poland instead of Equestria, but the resemblance is pretty striking). There are probably many similar examples in other works of culture, if you know any then let me know in the comments!
    The lighthouse is something that guides us, helps us find a way to a safe haven in the troubled times. It works so well for this movie, where Argyle seems to have the knowledge that no one else has. Interesting thing is that Twilight's cutie mark was used in the Zephyr Heights station, but only as a symbol of friendship between the three races. There is no depiction of any of the Mane 6 outside of Argyle's lighthouse, which could mean that the others just forgot about them (there are some theories involving a Memory Stone, because a drawing of it is present in the lighthouse too, among many other interesting artifacts and pictures). So it might be the only place in the whole world where the knowledge about the old times was kept.
    I just hope it will be rebuilt and we won't get a similar case as Twilight's library...
  22. Silly Druid
    This quote (I think you all know where it comes from) shows us that complex physical objects are all different from each other. Ranging from very big ones like galaxies, to very small like the aforementioned snowflakes, there are no two identical instances of them. But what happens if we consider really small objects, like atoms? Well, the atoms of one chemical element aren't actually all the same, but they are not all unique either. Every element has several versions called isotopes, which differ by the number of neutrons in the nucleus. So let's go deeper, to the most elementary particles that we currently know, like electrons and quarks. According to our current physical knowledge, the particles of a given type are all identical. It can mean one of three things:
    1. There are differences between them, but we can't detect them due to the technological limitations.
    2. They have a complex structure, but there is some physical process that makes them all the same.
    3. They are very simple objects that are indeed identical, which means we are close to discovering the fundamental structure of the universe.
    I don't know about you, but I think option 3 is the most likely one.
  23. Silly Druid

    From tip to TOE

    By Silly Druid,


    Physics
    By TOE I mean "Theory of Everything". The basic rules that govern our universe (so if you think there are multiple universes, it's not really about everything). There are several attempts to unify all physics, but there is no widely accepted version of it. I have an intuition what it should be, which I'm going to share with you here.
    I believe the fundamental theory is not geometric. Geometry should be an emergent phenomenon, a statistical property of a large number of basic objects, something like pressure or temperature. Human brain likes to think geometrically, that's why many theories are like this, including the string theory and its generalizations which have all these multi-dimensional objects called "branes". But I don't think it's the ultimate answer.
    So, if geometry is not fundamental enough, then what is? I think it's information. I can't think of anything more basic than that. In an earlier entry in this blog I stated that I don't think we live in a simulation, but actually the universe can be something similar to a computer with a very large number of bits of information in its memory. I don't know any theory like this, but if it exists, it may be close to the true Theory of Everything.
  24. Silly Druid

    Do Extra Dimensions Exist?

    By Silly Druid,


    Physics
    (Sorry for posting this 2 days later than usual.)
    In some physical theories there are not only the 4 dimensions we know (3 space and 1 time), but also some small extra ones. But they hasn't been observed yet, so do they really exist? Well, there is a very convincing argument (for me at least) that they do. It's called "CPT symmetry".
    Symmetry is a very important concept in physics, it means that some differences between two systems don't really affect the way they work. For example the "translation symmetry" means that the place where something is located is not important for its physical evolution (if all other conditions are the same).
    So what do these 3 letters mean? C is "charge symmetry", which means replacing all matter with anti-matter and vice versa. P is "parity", and means making a mirror image of the universe, and "T" is reversing the direction of time. The interesting thing is that individually all these symmetries are broken (not very strongly, but there are some subtle physical effects that don't obey them), but the combination of all three of them, as far as we know, holds. It also means that combinations of two of them are also broken, and are equivalent to the third one, for example CP = T.
    Without the extra dimensions, all this doesn't make much sense. The P symmetry is about space, T is about time, and C is about... some numbers we attribute to the particles to describe how they interact with each other. But, the extra dimensions that appear is some theories (starting with the Kaluza-Klein theory, and including some newer ones, like the string theory) are used to explain the charges as movement of the particles in these extra dimensions (which, due to their small size, is quantified, that's why the charges have discrete values).
    Making a mirror image (P symmetry) can be viewed as reversing all 3 spatial dimensions, T symmetry is reversing the time dimension, and if there are extra dimensions, then C symmetry is also reversing them. So, in other words, the CPT symmetry can be interpreted as "if you reverse all the dimensions, it's like you reversed none." So it all starts making sense, by grouping 3 similar symmetries into one. Coincidence? I think not.
  25. Silly Druid

    The Nature of Time

    By Silly Druid,


    Physics
    What is time? According to physics, it's just another dimension. There is a slight difference between spatial and time dimensions. Basically, the distance across two or more spatial dimensions is the square root of the sum of squares of the distances in the individual dimensions (as in Pythagorean theorem). But, due to time being a different kind of dimension, the distance across a spatial dimension and a time dimension is the square root of the difference of squares. Also there is only one time dimension, and three spatial dimensions. (There may be also some small extra dimensions, more about that in a future entry.)
    In Einstein's General Theory of Relativity the universe is a four-dimensional object, and the thing it's made of is called "spacetime". Quantum mechanics, especially my favorite interpretation of it, called many-worlds interpretation (another subject for a future blog entry) makes it more complicated, because the universe seems to "split" every time something random happens, so it becomes a very complex fractal-like object. Anyway, there is nothing in the current physical knowledge (that I'm aware of), that would suggest that the passage of time is a real thing. The universe as a mathematical object just exists, which suggests that the flow of time is an illusion, it's how we perceive it due to the way our brains work, but it doesn't really happen.
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