#hello everyone !!

[lovelyghst]

poor simon settling for just the tip with his pretty girl late at night, having just gotten home from work and finding her sleeping soundly in their bed. he gently coaxes you awake, going against every nerve in his worn out body to let you rest, and he asks you so, so nicely.

begs, more like it, and you simply can’t refuse.

with your sweet permission, he slicks up his cockhead and eases it into your little hole with some effort from the both of you, his eyes fluttering shut as he fights to maintain his promise. just the tip.

and he shows such good restraint for you! moaning through his clenched jaw as he slowly fucks the tip of his cock in and out of your pretty pussy, whining when the exposed length of his dick pulses from neglect. he’s sat back on his knees and heels with your hips pulled into his lap, not trusting his tired muscles enough for missionary. still, he can’t keep his hands off of you.

he may be desperate, but he couldn’t bring himself to hurt or force anything onto his princess. that would be the true death of simon riley. he even runs a gentle thumb over your swollen clit to make you melt into the pillows, urge those lovely little noises from your lips, the same ones he’s been hearing in his daydreams while he’s at work.

they make his dick throb, the seasoned soldier’s hand trembling as it soothes over your lower tummy. gosh, he missed you so much.

and you read it all on his face; how much he respects your wishes, but also how badly he needs relief. the slackened jaw, panting chest, droopy eyes heavy beneath furrowed brows. it makes you frown.

“simon,” you whine out softly, and his eyes snap up to meet yours. the look on your face makes him stifle a choked moan. “c’mere…”

you reach up as he leans forward for you and, to his surprise, you tug him in by his neck for a needy kiss. you wrap your legs around his lower back as best you can, locking them tight in the divot of pure muscle, and you reel him in closer.

consequently, the rest of his cock fully sheathes inside you and the sudden stretch makes you whimper out, him groaning loudly like a whore as he buries his heated face in your collar. christ, he fucking came just from one stroke of tight, wet warmth. and it feels so good, too good for him. he works hard, you think, he deserves it.

you giggle as you hold your baby close, let him catch his breath and grasp his settings before he rolls over, you held clasped in his arms.

he falls asleep in the matter of seconds, with his face in your chest and his dick in your cunt, like a good soldier. probably mutters some strained apology in between, even though you couldn’t be happier.

[evevsy]

after dark ✧

[daniilune]

You have come here in pursuit of your deepest urge... In pursuit of that wish which till now has been silent... silent...

~ 🖤

[physicallyimprobable]

what's the 3-dimensional number thing?

Well I'm glad you asked! For those confused, this is referring to my claim that "my favorite multiplication equation is 3 × 5 = 15 because it's the reason you can't make a three-dimensional number system" from back in this post. Now, this is gonna be a bit of a journey, so buckle up.

Part One: Numbers in Space

First of all, what do I mean by a three-dimensional number system? We say that the complex numbers are two-dimensional, and that the quaternions are four-dimensional, but what do we mean by these things? There's a few potential answers to this question, but for our purposes we'll take the following narrative:

Complex numbers can be written in the form (a+bi), where a and b are real numbers. For the variable-averse, this just means we have things like (3+6i) and (5-2i) and (-8+3i). Some amount of "units" (that is, ones), and some amount of i's.

Most people are happy to stop here and say "well, there's two numbers that you're using, so that's two dimensions, ho hum". I think that's underselling it, though, since there's something nontrivial and super cool happening here. See, each complex number has an "absolute value", which is its distance from zero. If you imagine "3+6i" to mean "three meters East and six meters North", then the distance to that point will be 6.708 meters. We say the absolute value of (3+6i), which is written like |3+6i|, is equal to 6.708. Similarly, interpreting "5-2i" to mean "five meters East and two meters South" we get that |5-2i| = 5.385.

The neat thing about this is that absolute values multiply really nicely. For example, the two numbers above multiply to give (3+6i) × (5-2i) = (27+24i) which has a length of 36.124. What's impressive is that this length is the product of our original lengths: 36.124 = 6.708 × 5.385. (Okay technically this is not true due to rounding but for the full values it is true.)

This is what we're going to say is necessary to for a number system to accurately represent a space. You need the numbers to have lengths corresponding to actual lengths in space, and you need those lengths to be "multiplicative", which just means it does the thing we just saw. (That is, when you multiply two numbers, their lengths are multiplied as well.)

There's still of course the question of what "actual lengths in space" means, but we can just use the usual Euclidean method of measurement. So, |3+6i| = √(3²+6²) and |5-2i| = √(5²+2²). This extends directly to the quaternions, which are written as (a+bi+cj+dk) for real numbers a, b, c, d. (Don't worry about what j and k mean if you don't know; it turns out not to really matter here.) The length of the quaternion 4+3i-7j+4k can be calculated like |4+3i-7j+4k| = √(4²+3²+7²+4²) = 9.486 and similarly for other points in "four-dimensional space". These are the kinds of number systems we're looking for.

[To be explicit, for those who know the words: What we are looking for is a vector algebra over the real numbers with a prescribed basis under which the Euclidean norm is multiplicative and the integer lattice forms a subring.]

Part Two: Sums of Squares

Now for something completely different. Have you ever thought about which numbers are the sum of two perfect squares? Thirteen works, for example, since 13 = 3² + 2². So does thirty-two, since 32 = 4² + 4². The squares themselves also work, since zero exists: 49 = 7² + 0². But there are some numbers, like three and six, which can't be written as a sum of two squares no matter how hard you try. (It's pretty easy to check this yourself; there aren't too many possibilities.)

Are there any patterns to which numbers are a sum of two squares and which are not? Yeah, loads. We're going to look at a particularly interesting one: Let's say a number is "S2" if it's a sum of two squares. (This thing where you just kinda invent new terminology for your situation is common in math. "S2" should be thought of as an adjective, like "orange" or "alphabetical".) Then here's the neat thing: If two numbers are S2 then their product is S2 as well.

Let's see a few small examples. We have 2 = 1² + 1², so we say that 2 is S2. Similarly 4 = 2² + 0² is S2. Then 2 × 4, that is to say, 8, should be S2 as well. Indeed, 8 = 2² + 2².

Another, slightly less trivial example. We've seen that 13 and 32 are both S2. Then their product, 416, should also be S2. Lo and behold, 416 = 20² + 4², so indeed it is S2.

How do we know this will always work? The simplest way, as long as you've already internalized the bit from Part 1 about absolute values, is to think about the norms of complex numbers. A norm is, quite simply, the square of the corresponding distance. (Okay yes it can also mean different things in other contexts, but for our purposes that's what a norm is.) The norm is written with double bars, so ‖3+6i‖ = 45 and ‖5-2i‖ = 29 and ‖4+3i-7j+4k‖ = 90.

One thing to notice is that if your starting numbers are whole numbers then the norm will also be a whole number. In fact, because of how we've defined lengths, the norm is just the sum of the squares of the real-number bits. So, any S2 number can be turned into a norm of a complex number: 13 can be written as ‖3+2i‖, 32 can be written as ‖4+4i‖, and 49 can be written as ‖7+0i‖.

The other thing to notice is that, since the absolute value is multiplicative, the norm is also multiplicative. That is to say, for example, ‖(3+6i) × (5-2i)‖ = ‖3+6i‖ × ‖5-2i‖. It's pretty simple to prove that this will work with any numbers you choose.

But lo, gaze upon what happens when we combine these two facts together! Consider the two S2 values 13 and 32 from before. Because of the first fact, we can write the product 13 × 32 in terms of norms: 13 × 32 = ‖3+2i‖ × ‖4+4i‖. So far so good. Then, using the second fact, we can pull the product into the norms: ‖3+2i‖ × ‖4+4i‖ = ‖(3+2i) × (4+4i)‖. Huzzah! Now, if we write out the multiplication as (3+2i) × (4+4i) = (4+20i), we can get a more natural looking norm equation: ‖3+2i‖ × ‖4+4i‖ = ‖4+20i‖ and finally, all we need to do is evaluate the norms to get our product! (3² + 2²) × (4² + 4²) = (4² + 20²)

The cool thing is that this works no matter what your starting numbers are. 218 = 13² + 7² and 292 = 16² + 6², so we can follow the chain to get 218 × 292 = ‖13+7i‖ × ‖16+6i‖ = ‖(13+7i) × (16+6i)‖ = ‖166+190i‖ = 166² + 190² and indeed you can check that both extremes are equal to 63,656. No matter which two S2 numbers you start with, if you know the squares that make them up, you can use this process to find squares that add to their product. That is to say, the product of two S2 numbers is S2.

Part Four: Why do we skip three?

Now we have all the ingredients we need for our cute little proof soup! First, let's hop to the quaternions and their norm. As you should hopefully remember, quaternions have four terms (some number of units, some number of i's, some number of j's, and some number of k's), so a quaternion norm will be a sum of four squares. For example, ‖4+3i-7j+4k‖ = 90 means 90 = 4² + 3² + 7² + 4².

Since we referred to sums of two squares as S2, let's say the sums of four squares are S4. 90 is S4 because it can be written as we did above. Similarly, 7 is S4 because 7 = 2² + 1² + 1² + 1², and 22 is S4 because 22 = 4² + 2² + 1² + 1². We are of course still allowed to use zeros; 6 = 2² + 1² + 1² + 0² is S4, as is our friend 13 = 3² + 2² + 0² + 0².

The same fact from the S2 numbers still applies here: since 7 is S4 and 6 is S4, we know that 42 (the product of 7 and 6) is S4. Indeed, after a bit of fiddling I've found that 42 = 6² + 4² + 1² + 1². I don't need to do that fiddling, however, if I happen to be able to calculate quaternions! All I need to do is follow the chain, just like before: 7 × 6 = ‖2+i+j+k‖ × ‖2+i+j‖ = ‖(2+i+j+k) × (2+i+j)‖ = ‖2+3i+5j+2k‖ = 2² + 3² + 5² + 2². This is a different solution than the one I found earlier, but that's fine! As long as there's even one solution, 42 will be S4. Using the same logic, it should be clear that the product of any two S4 numbers is an S4 number.

Now, what goes wrong with three dimensions? Well, as you might have guessed, it has to do with S3 numbers, that is, numbers which can be written as a sum of three squares. If we had any three-dimensional number system, we'd be able to use the strategy we're now familiar with to prove that any product of S3 numbers is an S3 number. This would be fine, except, well…

3 × 5 = 15.

Why is this bad? See, 3 = 1² + 1² + 1² and 5 = 2² + 1² + 0², so both 3 and 5 are S3. However, you can check without too much trouble that 15 is not S3; no matter how hard you try, you can't write 15 as a sum of three squares.

And, well, that's it. The bucket has been kicked, the nails are in the coffin. You cannot make a three-dimensional number system with the kind of nice norm that the complex numbers and quaternions have. Even if someone comes to you excitedly, claiming to have figured it out, you can just toss them through these steps: • First, ask what the basis is. Complex numbers use 1 and i; quaternions use 1, i, j, and k. Let's say they answer with p, q, and r. • Second, ask them to multiply (p+q+r) by (2p+q). • Finally, well. If their system works, the resulting number should give you three numbers whose squares add to 15. Since that can't happen, you've shown that the norm is not actually multiplicative; their system doesn't capture the geometry of three dimensions.

[moogle-woogle]

Monday Bank Holiday, so I made a couple of hats for Haruka

(Alt text included)

[okitanoniisan]

[strawbsonmars]

[greatandquestionablecontent]

An introduction to narrative styles of the Locked Tomb series, care of me rambling about it on discord

Book 1 is like LMAO BONES. And hot girls 😏 Murder mystery dinner party, with sciency magic but mainly we just understand the swords. Protag has no idea what’s going on but that’s actually very helpful for helping the reader understand what’s going on.

Book 2 is like Psychedelic Celestial Griefcore, with very complex magic Sci Fi that gets elaborated on at length. Dark. We are In Space for most of the book. God is there. Protag has no idea what’s going on because her brain is fucked and it’s… sad? And extra confusing for the reader.

Book 3 is like Sunshine and Rainbows! Too much sunshine for most people actually! All the surrounding characters are living in a shitty urban sci fi dystopia but the narrator is full of love and joy in her city of riots and poverty and sweltering sun. Protag no idea what’s going on because she’s only been alive for 6 months. Love & Light 🥰

[sculkshrieking]

teasing

[americiumam]

birthday gift for my bestie @averythebean :D

[tavs-brainworm]

Ascended Astarion talks a big game but the second you whip out the praise it’s all over. Stroke his ears and tell him how devoted you are to him and how happy and safe he makes you feel and he’s coming untouched before you can even say good boy

[funnywizard3000]

yay bunny suit loop yay yay

[dreamingonmyown]

they really gave us meredith “mommy issues” grey and addison “good girl” montgomery and thought we wouldn’t run wild

[niymue]

yee—haw

[cellarspider]

Good lord I take a week off of tumblr and now there’s a lot of you

Hello to everyone who’s just followed me in the past week! Most of you have come from a long ramble of mine on interdisciplinary learning, medieval head trauma, and Gallus’ well-wishings on my recent graduation (https://www.tumblr.com/gallusrostromegalus/727017193756327936), thank you to Gallus for that. Thank you to those of you who’ve commented with kind words as well. Specific shout-outs, links to relevant rambles, and questions are below, in the section “Link Roundup and Shoutouts”.

Yes, this is a post with sections. This is how we roll here.

Introduction to Spider

For those who don’t know, I’m Spider! I’ve just gotten my PhD in Mammalian Genetics, having gotten a Masters in Informatics and a Bachelors in Medieval Studies before that. I’ll quite happily ramble about any of them, with the following caveats: an undergraduate degree means I know the basics, but they may be increasingly out of date. And advanced degrees are increasingly specialized in their scope as you go along—you gain the skills to more easily understand things from related specialties, but you only become truly, deeply knowledgeable on very specific topics. However, these topics are not always limited to the field of study generally expected by the degree-granting institution! My focus ended up being significantly divergent from everyone else’s, which resulted in an interesting challenge of communicating my project to others at the institute.

The field I dove into for my PhD was systems genetics. Rather than studying individual genes and how they function, my work examined the wider view: think the difference between a local weather forecast versus modeling the global climate. Both synthesize vast amounts of information, just on different scales and levels of detail.

Many people love studying the tiny details around individual genes, because they can dig down into the mechanisms that make the gene work, how it might break and cause disease, and maybe how to fix those diseases. My love is for the global view of things, which gives you the ability to characterize general statements about how genes are regulated and modified. It’s a field that’s very hard to study without good data that’s complicated to acquire, so it’s a very exciting subject to work on! I’m looking forward to carrying that on into a postdoctoral study, in which I’ll work with a new lab and learn the dreaded skill of grant writing. I’ll be starting this month!

…As Gallus mentioned, my time until then is very much devoted to Baldur’s Gate 3. Happily for me, the new research group I’ll be joining has also been going nuts for Baldur’s Gate 3, so I’ll have a lot to talk about with my coworkers once I’m back to the lab.

In my free time, I’m happy to ramble upon request about the subjects I love, including but not limited to my fields of academic study, my constructed language hobby, scientific ethics and its portrayal in media, creepy-crawlies (always appropriately tagged for people’s phobias), and Baldur’s Gate 3.

…Lots of Baldur’s Gate 3. (I’ve only just reached the Lost Light Inn, please no spoilers!)

Link Roundup and Shoutouts

For those who are interested to see my ramble about why European medical texts in the medieval period tended to be terrible, it’s available here: https://www.tumblr.com/cellarspider/680342023316930560/hi-please-rant-about-medieval-european-medical

Thank you to all those who dug up the name of the academic text I’d forgotten! Its title, in all its wordy glory, is Injuries of the skull and brain, as described in the myths, legends, and folk-tales of the various peoples of the world, with some comments on the significance and reliability of this information in evaluating contemporary concepts as to their nature and lethality by Cyril B. Courville, 1967. It’s a fantastic book, and good lord that title just does not stop

Thank you to fellow spiders @one-spider-from-mars and @vaspider for their comments. We are many. We are mighty.

Thank you to @belovedbright for the fantastic story of the death of Conchobar mac Nessa via brain trauma inflicted by a brain https://www.tumblr.com/belovedbright/727132485919604736

To @doomhamster's question on whether egg whites were used in the medieval treatment of burns: I don’t know! Unfortunately I can’t access the translation of the medical manual I referred to back then (https://worldcat.org/title/1123716578), and the only version I can find online at the moment is in 14th century French (https://www.bl.uk/manuscripts/FullDisplay.aspx?ref=Sloane_MS_1977). Egg whites do appear 33 times in the translation, according to the limited ability I have to search the text, and they show up throughout the book.

[marrowsucker]

ꕀ i don’t have a name cause im not a person ꕀ

ꕀ you can just call me whatever you like ꕀ

ꕀ for example you could call me jōb or lewis ꕀ

ꕀ im a they ꕀ

ꕀ im 22 and i got autism and i love sturgeon ꕀ

ꕀ my main acc is @sturgeonposting ꕀ

𐂯 this is where I intend to post about my 𐂯

𐂯 original characters 𐂯

𐂯 hyperfixations 𐂯

𐂯 silly life 𐂯

𓆝 𓆟 i have a lot to say and none of it is good 𓆟 𓆞