I was reading about fractals and the book said a fractal is an object whose fractal dimension exceeds its topological dimension.

But I do not understand why a koch snowflake has a topological dimension of 1 and not 2, like any other shapes with closed loops like a square, circle etc.

I tried reading the wiki but didn't understand any of it. Can anyone explain this to me like I am 5?

PS: Sorry if its a stupid question

Here's the link to the article

> Kovner and Rosenstein. > Int. J. Mod. Phys. A 07, 7419 (1992). > Topological Interpretation of Electrical Charge, Duality and Confinement in 2+1 Dimensions.

http://www.worldscientific.com/doi/abs/10.1142/S0217751X92003392

Thanks!

I do wonder whether it still make sense to write the bibliographic details here...

Topology in 1D, 2D and 3D makes perfect sense, but how does it work in 4D and above? Do holes get different properties? Does this question even make sense?

Is there an intuitive concept of how dimension operates in incidence posets of graphs?

For the life of me I cannot quite understand what is meant by the set of total order of vertices, whose combination yields the partial order. If the partial order < is defined between vertices and edges, (a < b iff a is a vertex, b is an edge, and a is one of b's endpoints), then how do we define the total order among vertices? Am I missing something here?

Suppose for an recursive equation: T(n) = a*T(n/b) + cn Could we chart each iteration as an ath dimensional polyhedron, with composition described as cn?

My TA was talking about the history of the Poincaré Conjecture, and explained how first it was proved for dimensions 5 and up, then dimension 4, and finally dimension 3. He explained that in Topology, dimensions 4 and 3 were harder to work with. Can anyone explain why? I only have a very basic knowledge of Topology, but am intrigued as to why dimensions 3 and 4 are different from the rest.

This recurring thread will be for questions that might not warrant their own thread. We would like to see more conceptual-based questions posted in this thread, rather than "what is the answer to this problem?". For example, here are some kinds of questions that we'd like to see in this thread:

• Can someone explain the concept of maпifolds to me?
• What are the applications of Represeпtation Theory?
• What's a good starter book for Numerical Aпalysis?
• What can I do to prepare for college/grad school/getting a job?

Including a brief description of your mathematical background and the context for your question can help others give you an appropriate answer. For example consider which subject your question is related to, or the things you already know or have tried.

Hi everyone, I have been playing around with the idea that a 1 x 1 square could be compressed into the 1st dimension only by stretching its length to infinity to achieve a 1/infinitely thin line and maintain the information. I have been trying to figure out if this is true or not mathematically but haven't been able to come up with an equation that would also work in higher dimensional "compressions". Any help would be great thanks. Also I do realize that I'm implying infinity over infinity is equal to one in this scenario, because I think this case would not end up in infinity/infinity being undefined. Is this true? Does it go up to higher spatial dimensions? Thanks a lot, hope I'm not just idiotically wasting your time!

So I just feel like it's time I broke down what a lot of the symbolism and metaphorical things in Destiny 2 are loosely based on, well maybe more then just loosely based on. I've broken it down into two basic schools of thought or two philosophies that kind of merged over the centuries and became Alchemy.

So since there's so many alchemical symbols related to witch queen I thought I'd make a post trying to explain some of it. Warning this is going to be a longer post and I'm not going to make a TRDL because to understand this you have to ready to take in a bunch of this information, and most people just aren't ready this or as I'm going to refer to it as "it"

-Now before I can get into how these philosophies are related to bungie as a company and also related to Destiny 2s inspiration for themes and symbols I need to explain things as best as I can and also I need to share my perspective of why I see things the way I do.

First of these two things that lead into alchemy is Sacred Geometry. I think it's obvious because sacred geometric symbols are just pouring out of every Bungie game Destiny, Halo, Pathways to darkness and Marathon. Slowly over time slowly over time bungies has been getting more and more liberal with the symbolism of sacred geometry and having it placed right out in the open. I mean you can just see it all over the dreaming city and the destinations menu. Here's a good example

https://imgur.com/lR5dquN

Thats a screen shot I took in the dreaming city. On the bottom left is the flower of life, and on the right is Metatron's cube if you want go ahead and look at an image of the two of those stacked together it's pretty obvious what this art work in the dreaming city has been inspired by.

-The second part is bungies inspiration from Hermiticism or Hermitics. Hermitics is something most people, even the wiki page itself don't really understand. Here's a decent site that breaks down Hermitics a bit better then the normal wiki page:

https://www.newworldencyclopedia.org/entry/Hermeticism

So to explain what Hermitics is. It's not really a religion, at least in todays age. It's a philosophy which is tied to the major themes and symbolism found trough out almost every single major religion in entire world. These themes and symbols fly over most peoples heads and go unnoticed and if they are noticed they're normally misunderstood.

I guess the be

An idea that's recently crystalized on battleboards is that some things are more than infinite or "beyond infinity." This is usually attatched to a poor attempt of abstracting something that's already infinite (or supposedly infinite) into something "greater."

First and foremost, infinite simply means not finite. In other words, if something isn't infinite then it's finite.

Excuses are rebuttals

The two excuses used to justify this is either A) dimensional tiering, and B) transfinite numbers.

So let's address these.

A) Dimensions have nothing to do with infinity.

Dimensions is a property of a space (topological space, vector space, etc.). In other words, dimensions can't exist without a space.

A space can be either discrete or a continuum. A discrete space is a space with a minimal, nonzero displacement. A continious space where any displacement (arbitrary infinite sequence past the decimal point) is allowed.

E.g. discrete spaces: ℕ^(n) (natural numbers), ℤ^(n) (integers).

E.g. continious spaces: ℝ^(n) (real numbers), ℂ^(n) (complex numbers).

We're going to use ℤ^(n) and ℝ^(n) for the demonstration.

The n denotes the number of dimensions, e.g. ℝ^(3) = ℝ × ℝ × ℝ (each ℝ representing a perpendicular direction with the given coordinates x,y,z) is a three-dimensional space.

Similarly ℝ^(5) = ℝ × ℝ × ℝ × ℝ × ℝ, and where an arbitrary point in this space is given by the coordinates (x,y,z,u,v) where each of these coordinates is given by a real number.

So discrete spaces are countably infinite, and continious spaces are uncountably infinite. This is because the set of naturals and integers are countable and the set of reals and complex numbers are uncountable.

So it's true that |ℝ| > |ℤ| (where |x| denotes the cardinality [size] of the set x).

Now the "VSBW idea" is that |ℝ^(3)| > |ℝ| because one space has more dimensions...this is demonstrably wrong. Fact is that |ℝ^(m)| = |ℝ^(n)| for all natural numbers m, n > 0, similarly |ℤ^(m)| = |ℤ^(n)|.

Similar to how ∞ = ∞ + 1 = ∞ ⋅ 2 = ∞^(2), this is just how infinity works. ∞ + 1 might seem larger than just ∞ (after all x + 1 > x for all finite numbers), but it's really not.

In other words a one-dimensional space has the same cardinality as a gogolplex-dimensional space.

So more dimensions do not make you "more infinte" (let alone "more

"Staples" are good cards you can play in a variety of decks. They can't all be played in just any deck, but they are ones you should at least consider. I have sorted them into categories, and within each category they are sorted from lowest rarity to highest rarity. If you have any suggestions for additions, please let me know in the comments below!

Link 1 Monsters

• Gravity Controller (R)
• Secure Gardna (R)
• Link Spider (SR)
• Linkuriboh (UR)
• Relinquished Anima (UR)
• Salamangreat Almiraj (UR)

Link 2 Monsters

• Pentestag (N)
• Cross-Sheep (R)
• Artifact Dagda (SR): Requires Artifact Scythe (UR)
• Barricadeborg Blocker (SR)
• Knightmare Cerberus (SR)
• Knightmare Phoenix (SR)
• PSY-Framelord Lambda (SR)
• Crystron Halqifibrax (UR)
• I:P Masquerena (UR)
• Predaplant Verte Anaconda (UR)

Link 3 Monsters

• Hraesvelgr, the Desperate Doom Eagle (R)
• Tri-Gate Wizard (SR)
• Curious, the Lightsworn Dominion (UR)
• Knightmare Unicorn (UR)
• Mecha Phantom Beast Auroradon (UR)
• Ningirsu the World Chalice Warrior (UR)
• Topologic Trisbaena (UR)

Link 4 Monsters

• Topologic Bomber Dragon (SR)
• Accesscode Talker (UR)
• Apollousa, Bow of the Goddess (UR)
• Borreload Dragon (UR)
• Borrelsword Dragon (UR)
• Mekk-Knight Crusadia Avramax (UR)
• Saryuja Skull Dread (UR)
• Topologic Zeroboros (UR)

Link 5 Monsters

• Underworld Goddess of the Closed World (UR)

Xyz Monsters

• Evilswarm Nightmare (R)
• Downerd Magician (SR)
• Number 60: Dugares the Timeless (SR)
• Number 39: Utopia Double (SR): Requires Number 39: Utopia (UR) and Double Or Nothing! (N)
• Number 41: Bagooska the Terribly Tired Tapir (SR)
• Number F0: Utopic Future (SR)
• Number F0: Utopic Draco Future (UR)
• Abyss Dweller (UR)
• Beatrice, Lady of the Eternal (UR)
• Divine Arsenal AA-ZEUS - Sky Thunder (UR)
• Evilswarm Exciton Knight (UR)
• Toadally Awesome (UR)
• Tornado Dragon (UR)

Fusion Monsters

Summon with Super Polymerization:

• Diplexer Chimera (N)
• World Chalice Guardragon Almarduke (R)
• Starving Venom Fusion Dragon (SR)
• Predaplant Dragostapelia (SR)
• Mudragon of the Swamp (SR)

Summon with Instant Fusion:

• Thousand-Eyes Restrict (SR)
• Millennium-Eyes Restrict (UR)

Sent from Extra Deck to GY, e.g. by Dogmatika Punishment:

• Elder Entity N'tss (UR):

Synchro Monsters

• Stardust Charge Warrior (R)
• Coral Dragon (SR)
• Formula Synchron (SR)
• Herald of the Arc Light (SR)
• Martial Metal Marcher (SR)
• Borreload Savage Dragon (UR)
• Chaos

I was just fifteen years old when I learned monsters were real.

That day, a Tuesday, I recall, I was a little later than usual coming home from school, on account of joining the Science Club. I’d just recently watched Donnie Darko for the first time, and had become enthralled with the idea of time travel. As I walked home, backpack weighing me down, I realized I was going to miss the start of my favorite documentary series, and had to do something drastic if I intended to change that.

There was a shortcut that ran through one of the yards in the neighborhood, but I rarely used it for fear of being caught. The old man who lived there was generally belligerent, and if he caught anyone cutting through his property he’d yell and chase them away, threatening to get his gun. No one had actually seen his gun, mind you, but no one wanted to, either. Perhaps I was feeling brave, or the thought of missing my favorite show was too much, but that day I decided the time I’d save was worth the risk.

After jumping the old fence, I made my way along the side of the house and into the backyard. I cursed myself for wearing my Triforce hat and orange vest, as high visibility an outfit as one could find. I was about halfway across the yard when I heard a loud splash behind me, like someone jumping off a high board. I vaguely remembered the old man having an above ground pool which he likely never used, letting the water fester and bloom. The idea of old man Williams splashing around in that fetid water was both ridiculous and disgusting.

And yet, something was in the pool. I watched the dirty water roil and churn, waves of it flowing over the sides. It looked as if an animal were drowning, and I stood frozen to the spot, not knowing whether I should run away from a place I shouldn’t have been in the first place, or run forward and help it. Time seemed to be rushing forward anxiously, the late-day sun arcing toward the horizon.

The sight of the writhing thing that clawed its way out of the pool changed me forever. One look at its twisted formation of limbs and bones and organ, familiar things twisted into new designs, murdered my innocence in an instant. Its grotesque face, with bloodshot eyes nearly popping out of its broken skull, fixed on me in one, chilling instant.

And then it was chasing me, bones popping and cracking, shuffling and rearranging its hideous form. And it screamed, too, screamed a single sound at me, a word like, “Nha!” The voice bloody and raw, the

Good morning. I am interested in learning more about N-dimensional geometry. Is there a beginner's course available online or a book that introduces the concepts?

I have an undergraduate in mathematics, but bring very little other knowledge to the table. What would such a course even be called?

Hey everyone, you might remember me as that one person that writes up TOO MANY POSTS about Weather Painters and after about 2 years I'm glad to be back thanks to last night's reveal for the upcoming main set Dimension Force.

In case anyone is out of the loop here, this is what was revealed last night:

> Tenki Yohou / The Weather Forecast

> Field Spell Card

> You can only activate 1 card with this card’s name per turn. You can only use this card name’s (2) and (3) effects once per turn each.

> (1) When this card resolves, you can place 1 “The Weather” Spell/Trap directly from your Deck to your Spell & Trap Zone face-up.

> (2) You can also treat face-up “The Weather” cards in your Spell & Trap Zones as “The Weather” monsters and use them for the Link Summon of a “The Weather” Link Monster.

> (3) During your Main Phase, you can; immediately after this effect resolves, Normal Summon 1 “The Weather” monster.

Translation provided by the lovely people at the YGOrganization.

Let's unpack how much this single card does for the deck:

• Consistency and playing through interruptions: While the deck has gotten several consistency tools over the years in the forms of Pot of Prosperity and the recent addition of Piri Reis Map it has always been very frail and has faced the issue of being overly reliant on getting to resolve Snow on the first turn. Either you get a hand that's good enough to set up a painter with 1 or 2 canvases PLUS 1 or 2 disruptions in the form of handtraps, floodgates and powerful backrow or you get a nigh unplayable mix of too many interruptions or too many starters. With Forecast we now have a play that starts, extends and is able to bait interruptions when going second all in one. Similar to Snow, Forecast places a card face up in the S/T zone which means it can't be hit by Ash and by setting up a canvas like Snowy or even Rainbowed before committing your normal summon you can ensure that the painter you play will be safe from handtraps or other disruptions thanks to the quick effect it's getting and being able to tag out. AND EVEN THEN, if your first norm

In recent decades the creation and development of coordination polymers has become an emerging topic because of their intriguing network topologies as well as their possible applications across a range of fields due to their magnetic, optical and nonlinear optical properties (such as nonlinear optical thermometry, for details see paragraphs below), and electronic features.In fact, the lanthanide-based coordination polymers are rapidly developing field due to their distinctive light emission properties which are beneficial for future applications. In general it can be stated that porous coordination polymers attracted a lot of attention over the last few years due to their diverse structures, large pores that can be tunable and appealing properties.However, the majority of scientific papers are focused on transition metals that typically have 6-coordinated or 4-coordinated, or f-block metal ions because of their distinctive properties in catalysis and fluorescence and catalysis, whereas CPs created by the main group metals like bismuth, which have one electron pair and are not often reported.

Because of its flexible geometric coordination environment, the zinc ion is considered to be an all-encompassing node in creation of coordination polymers.From the coordination chemistry viewpoint it is evident that the absence of a single pair of electrons influences the angle of coordination of the lanthanide ions.For instance the polyhedral shape of the europium ion is usually irregular, which differs from the common octahedron, tetrahedron d-block metal ions like iron.From the viewpoint of coordination number of typical lanthanide ion has many coordination numbers that result in the unpredictability of and variety in lanthanide-based structures.For instance it was reported earlier that an eight-coordinated europium and terbium-based CP built from basic construction units, shows an unprecedented level of topological complexity, with just one node that is unique.

Lanthanide ions feature naturally low absorption coefficients, which isrestricting their use in practical applications, which typically require high intensity of emission.This limitation is overcome by complexing the chromophore with organic fragments referred to as ‘antenna’ and ‘sensitizers’ to effectively absorb the visible light spectrum and transfer their energy into the states that are excited by central lanthanide and ions.Thus, selecting the right organic linkers is vital.In previous research was

I don't want to step on anybody's toes here, but the amount of non-dad jokes here in this subreddit really annoys me. First of all, dad jokes CAN be NSFW, it clearly says so in the sub rules. Secondly, it doesn't automatically make it a dad joke if it's from a conversation between you and your child. Most importantly, the jokes that your CHILDREN tell YOU are not dad jokes. The point of a dad joke is that it's so cheesy only a dad who's trying to be funny would make such a joke. That's it. They are stupid plays on words, lame puns and so on. There has to be a clever pun or wordplay for it to be considered a dad joke.

Again, to all the fellow dads, I apologise if I'm sounding too harsh. But I just needed to get it off my chest.

So, I recently became interested in division rings, and exploring what linear algebra can be done over a division ring instead of a field. I find the theory of modules over a division ring and quaternion Hilbert spaces super cool, and I want to hear if anyone knows any practical applications to these ideas.

As some background, a division ring is a field minus the commutativity axiom. Surprisingly, quite a bit linear algebra still follows through when considering modules over division ring instead of modules over a field (i.e. "vector spaces"). For example, over a fixed division ring: every module has a well-defined dimension, and maps between modules satisfy the rank-nullity theorem.

Things get even more interesting when you consider modules over the quaternions in particular. Since the quaternions come with a canonical topology and norm, you can form analogues of topological vector spaces and Banach spaces over the quaternions. I had previously only known about real and complex Banach spaces, so it blew my mind when I discovered that you can also have quaternion Banach spaces! (Unfortunately, the space of maps between quaternion Banach spaces is only a real Banach spaces, but that's the only annoying quirk that I've come across). In fact, you can define an inner product over the quaternions in the same way you can define an inner product over the complex numbers, and then get Hilbert spaces over the quaternions! I could go on, but hopefully you can see how excited I am about quaternion vector spaces right now.

Unfortunately, however cool quaternions are, it appears quaternion vector spaces don't seem to get much use. Is this true, or have any cool results come from quaternion vector spaces?