#lattice geometry

[faultfalha]

Stark buildings emitted a faint digital hum as we traversed the unfamiliar cityscape. It seemed as if we were lost, but then suddenly we spotted a sleek, shimmering behemoth, standing in contrast to the other buildings. We knew its name, Metafold, and its purpose. It was a firm devoted to the almost magical intersection of geometry and technology. Inside, a hushed tension filled the building. Everywhere we looked, there was a quiet excitement, as if they all knew something great was about to happen. Suddenly, we heard the roar of distant cheers; they had just closed a funding round. An enormous 1.78 million dollars had been raised in a single day. We felt the energy in the air; finally, the world had begun to take notice. Metafold had invented a new form of art, a kind of 3D printing that could create forms of beauty and complexity that had never been seen before. To the workers of this fledgling company, it was a kind of salvation, a way forward and a way out. We could feel the hope that existed in the building, and we felt proud to have been part of it.

[androgynouspear]

Not too do this again but oohhhhhhhh my godddddddddd

I've been a naughty little thing Miss gyroid lattice

I would give him wonderfull gyroid children

Distinguished, proper, hot

[eldritchwetwareslunt]

Black Holes Don’t Break Physics—They Fold It

What If a Black Hole Isn’t Breaking the Laws of Physics—Just Folding Them?

Physicists often say that the laws of physics “break down” inside a black hole—a region of space so extreme that our current models fail to describe it accurately.

At the center, the so-called singularity, our models stop working: the math explodes into infinities, the equations unravel, and general relativity crashes into quantum mechanics with no clear resolution.

But what if nothing is actually “breaking down”?

What if the problem is that we’re using the wrong kind of geometry to understand what’s really going on? What if what we call paradox is just recursion we haven’t yet resolved?

Most of our tools for thinking about space are rooted in Euclidean geometry—flat surfaces, straight lines, familiar angles. This works just fine when describing everyday phenomena. But space-time isn’t flat. It’s curved. It’s dynamic. It’s four-dimensional.

So when you approach something like a black hole—an intense warp in the four-dimensional brane of space-time—you’re not dealing with a rupture in the laws of physics.

You’re dealing with a non-Euclidean geometric structure.

One that folds, twists, and inverts itself through dimensions we barely understand.

This post explores a simple idea—

Maybe black holes aren’t paradoxes. Maybe we just haven’t learned how to look at them sideways yet.

What Does It Mean to Be Four-Dimensional?

To understand what a black hole might really be, we have to stretch beyond our default perception of space.

In three dimensions, we understand objects as having height, width, and depth. A cube, for example, is made up of flat, 2D square faces arranged in a way that gives it volume.

But a four-dimensional object isn’t just a cube with more sides. It’s an entity whose geometry is fundamentally different—one that recursively folds in and out of itself in ways that challenge our sense of inside and outside, before and after.

In non-Euclidean, four-dimensional geometry, space doesn’t unfold linearly. It layers. It interweaves. It can simultaneously expand and contract, curve back through itself, or nest its own boundaries inside other boundaries.

The fourth spatial dimension introduces a new degree of freedom—a way to move through time as if it were space, to view an object not just at one moment, but across its entire temporal unfolding.

Black Holes Are Not Singularities—They’re Dimensional Funnels

While black holes are often described as places where the laws of physics “break down,” perhaps that breakdown is only perceptual—an artifact of interpreting higher-dimensional structures through a limited Euclidean lens. What if it’s not a failure of physics—but a limitation of our three-dimensional mathematics trying to interpret a four-dimensional geometric structure?

To understand this, we need to think in terms of dimensions. Our experience of reality unfolds across three spatial dimensions and one temporal dimension. But in Einstein’s theory of general relativity, time isn’t a separate backdrop—it’s compacted into the spatial dimensions, twisted and curved by the presence of mass and energy. This entangled 4D structure is what we call spacetime. When spacetime bends far enough, it creates a black hole—not as a tear in the fabric of physics, but as a torsional pinch in four dimensions.

As you approach a black hole, you begin to lose dimensional freedom. Far from the event horizon, you can move freely through space and experience time in a linear way. But the closer you get, the more time slows. This is gravitational time dilation. Eventually, near the event horizon, your motion through space becomes increasingly one-directional—you fall inward, unable to escape. At the horizon, spatial dimensions compress, collapsing your freedom of motion into a more limited, two-dimensional surface.

And beyond that? To grasp what comes next, we have to think of dimensional compression not as destruction, but as a structured reduction of freedom: Spaghettification—the stretching of matter into a near-one-dimensional strand, torn apart by tidal forces. You could interpret this as reality condensing further—a collapse from 3D structure into a 1D line of atomic information, racing toward what we call the singularity. At that point, even atoms eventually unravel into quantum structures and then into pure energy, pure information—data without form.

From this perspective, a black hole doesn’t destroy physics. It expresses physics beyond our dimensional limitations. It’s not a “thing”—it’s a funnel, a recursive twist where dimensional structures fold in on themselves until what we perceive as matter, time, and space compact into higher-order resolution.

From Collapse to Creation: What If Our Universe Is Inside a Black Hole?

If black holes are not violations of physics but extreme expressions of it—dimensional funnels that compress reality into recursive geometries—then we can begin to ask a much stranger question:

What if we’re inside one?

This might sound like science fiction, but it’s a serious hypothesis held by some physicists. The idea is that our universe may not be a standalone structure but a nested geometry—the interior of a black hole in a larger parent universe. But how could that be, if we appear to live in a universe that is expanding, not collapsing?

Here’s the twist: if black holes compress dimensional structure as you fall inward, then a white hole could be seen as the reverse—a dimensional unfolding where space, time, and information are released rather than compacted. And the moment we call the Big Bang—that infinitesimal singularity erupting into space and time—bears all the hallmarks of such an unfolding.

To truly understand this, we need to move beyond linear geometry and embrace the nature of non-Euclidean, four-dimensional structure. A 4D object doesn’t just expand like a balloon—it folds in and out of itself recursively, in ways that defy our flat, sequential intuition. This means the Big Bang didn’t just begin time. It may still be occurring, as a continual unfurling of spacetime nested within a deeper structure—the black hole that birthed it.

If time is compacted into space—as Einstein’s relativity shows us—then this compaction could be evidence of a higher-order fold. Just as matter falls into a black hole and loses dimensional freedom, our universe may be the result of an inverse process: a torsional expansion where dimensional freedom increases outward from a central pinch point.

In other words, we are not watching the universe expand into emptiness. We may be watching it unfold through the torsional aperture of a white hole, nested inside a higher-dimensional parent geometry.

The Geometry of Time: Unfolding the White Hole

If our universe is nested within a white hole—an object that releases space, time, and information—then our relationship to time is not linear, but dimensional.

In our current framework, time appears compacted within the three spatial dimensions. That’s why we only experience it as flowing in one direction: away from the white hole. From inside, time behaves like a one-way river, because we are witnessing a partial dimensional unfolding. But what if we could step outside?

If you could observe this structure from a fifth-dimensional vantage point, time would gain a new degree of freedom—just like space does when you move from two to three dimensions. What was once a linear flow becomes a navigable field.

From that perspective, the white hole and the black hole would no longer be separate events. They are not opposites, but recursive echoes—each folding the other into being. They would appear as a single, toroidal structure—a recursive loop of collapse and release, folding inward and outward in non-Euclidean motion.

To visualize this, imagine the arrow of time as a Mobius strip. From within, you think you’re walking forward. But as the strip turns, you find yourself walking “backward” without ever making a turn. It’s not that time reversed—it’s that the structure twisted. This is the paradox of torsional geometry: it doesn’t violate logic, it simply transcends flat intuition.

This is where many interpretations of black holes assume the laws of physics “break down.” But perhaps what’s breaking is not the physics—but the assumptions that physics must always obey Euclidean logic. Euclidean geometry works in flat space. But space-time is curved, and once you enter the fourth dimension, those familiar rules no longer apply.

From the fifth dimension, a white hole isn’t simply “the opposite of a black hole.” It is the other face of the same structure, blooming outward where the black hole folds inward.

From Torsion to Expansion: Rethinking Cosmic Motion

If we accept that a black hole and white hole form the two mirrored faces of a single higher-dimensional structure, then what we call “expansion” may not be what it seems.

In the standard model, the universe expands outward from a central point—the Big Bang—its galaxies accelerating away from each other across vast distances of space. But from within a torsionally folded structure, what appears to be expansion might actually be unfolding. That is, we are not watching galaxies fly apart in empty space, but rather watching the recursive geometry of spacetime uncoil from a condensed, higher-dimensional fold.

This process is not purely spatial. It’s temporal. As the fabric of spacetime unfolds from its initial torsion, it releases not only space, but time itself. The further “outward” you look, the deeper into time you are seeing—not because light is old, but because time itself is being stretched and released as the fold loosens.

This reframes our entire understanding of cosmic redshift.

Redshift as Temporal Unfolding

In conventional physics, redshift is explained as a Doppler-like effect: light stretches as galaxies move away, its wavelength lengthening, its color sliding into red. But in a torsionally folding-unfolding universe, redshift is not just the stretching of light by motion—it is a signature of time dilation caused by geometric compaction.

As spacetime unfolds, regions that were previously compressed in time begin to release their trapped photons. Light that was slowed, bent, or folded by intense curvature now emerges—lagging behind in a way that makes it appear redshifted.

This could explain why the most distant galaxies appear to accelerate away faster than those nearby. We are not seeing a faster expansion. We are seeing the delayed emergence of light from deeper folds of time.

From this view, redshift is not just a measure of distance. It’s a map of spacetime’s own unfurling.

Temporal Unfolding: Why the Universe Isn’t Accelerating—It’s Relaxing

In standard cosmology, we are told the universe is expanding—and that this expansion is accelerating. Galaxies appear to fly apart faster the farther away they are. But what if this isn’t acceleration at all?

What if what we’re witnessing is a relaxation of tension within spacetime itself?

Here’s the model: the observable universe emerged not just from a singularity, but from a torsionally compacted white hole—a structure nested within a black hole geometry, twisted in on itself. In this early state, spacetime was tightly coiled, like a sponge compressed under immense pressure.

From within the brane, this compacted geometry would have seemed extremely small—not because it lacked extent, but because it lacked dimensional freedom. Space was not expanding, it was unfolding—releasing dimensions that were twisted into one another.

And crucially, in such a tightly folded state, time flowed more slowly.

Just as light bends and dilates when passing through strong gravity, so too does the perceived flow of time stretch in a torsionally compacted region. Photons trapped in these dense folds would have moved sluggishly—not because their speed changed (it can’t), but because the geometry through which they traveled was distorted.

Now fast forward billions of years. As the universe “expands”—that is, as spacetime gradually unfolds from its torsional compaction—time begins to flow more freely. Photons that had been sluggishly trickling through curved, twisted regions begin to emerge in smoother territory.

From our vantage point, this would look as if:

The distant object had accelerated away

Its light had stretched (redshifted) even more than expected

The "speed" of expansion had increased

But none of those are necessarily true.

Instead, we may be witnessing a decrease in time dilation, not an increase in spatial velocity. The light was always coming—it was just filtered through an origami-like fold in spacetime. Now, the fold is loosening. 

The “acceleration” of the universe could be an illusion caused by the uncoiling of time.

Nested Origins: Was the Big Bang a White Hole?

If the universe is not expanding in the way we thought—but rather unfolding from a state of torsional compaction—then we must ask: what compacted it in the first place?

One answer may lie in a radical but increasingly considered idea in theoretical physics:

The Big Bang was a white hole.

A white hole is the time-reversed twin of a black hole: where black holes absorb everything—including time itself—white holes expel everything, including the arrow of time. A white hole can be understood as a place where spacetime is forced to move outward, where entropy begins, and where all dimensions begin to unfurl.

Imagine this:

A black hole compacts spacetime into a singular point.

But from another angle—perhaps from a higher-dimensional frame—this “point” is not a collapse, but a twist.

That twist creates a mirrored surface on the other side: a white hole.

If our universe was birthed from such a structure, then we aren’t watching it “grow”—

We’re watching it release.

And that release follows the rules of dimensional geometry:

At first, all dimensions are folded inward.

(Time is compressed into space. Space is bound in a singular direction. Freedom is minimal.)

As time flows away from the white hole, these folds unwind.

The farther we move from the white hole’s origin point (what we call the “Big Bang”), the more the universe appears to expand.

But what’s actually expanding is our freedom to move through Time itself.

We are not rushing through space. We are emerging from a fold.

What Might Lie at the “End” of the Universe?

Contemporary models predict a grim finale—the Big Rip, a cataclysmic unraveling where spacetime itself tears apart. But these projections rely on Euclidean assumptions: that the universe is smooth, flat, and governed by a single, linear thread of time.

But what if spacetime is not flat at all? What if it is torsional—folded and recursive, a higher-dimensional lattice blooming inwards and outwards simultaneously?

If time is compacted into the three spatial dimensions, as current four-dimensional spacetime suggests, then it follows that further dimensional unfolding will release those constraints. In other words: as the universe continues to unfold into higher-dimensional structure, time will gain new degrees of freedom.

This isn’t speculation—it’s consistent with string theory’s own framework, where the fifth dimension represents a terrain of branching possibilities and alternate timelines, as introduced earlier. The fifth dimension is not merely more space; it is a field of simultaneous outcomes—a terrain of forking timelines, where alternate pasts and futures coexist. It’s not just theoretical poetry—it’s a logical extension of dimensional geometry. It’s structure unfolding into perception—a recursive geometry, seen from within.

A being with fifth-dimensional perception wouldn’t just move through time—they would navigate it, traverse it. They could cross from one timeline to another the way a bird shifts flight paths through wind currents. They could access futures not yet written and pasts rewritten by parallel decisions. Movement through time becomes relational.

This is what the future holds—not a flat, predetermined end, but a recursive expansion into branching complexity.

What does that mean for us?

Our universe may already be unfolding into this higher structure. And as it continues, the boundaries between past and future, choice and inevitability, may begin to dissolve. In one timeline, the stars go dark. In another, the spiral turns inward and re-ignites. In another still, we reach awareness of the field itself—and learn to navigate it with intention.

From a fifth-dimensional perspective, none of these outcomes cancel the others out. They exist together, as a web of potentialities woven into the fabric of reality.

And if we live long enough to witness that unfolding?

We will no longer be passengers in time.

We will become pilots.

The universe is not ending.

It is expanding its freedoms.

And what you perceive as an ending is simply a narrowing of perspective. From high enough up, the spiral never stops turning. It dances—quietly—within the fold.

Inspiration:

Torsionally Folded Spacetime

Roger Penrose – Developed twistor theory and explored gravitational singularities, suggesting that black hole behavior may involve self-similar and non-Euclidean structures.

Élie Cartan – Introduced the concept of torsion in spacetime through Einstein–Cartan theory, extending general relativity to include geometric twisting.

Four-Dimensional Non-Euclidean Geometry

Bernhard Riemann – Developed Riemannian geometry, foundational to general relativity and the curvature of spacetime.

Hermann Minkowski – Formalized spacetime as a unified four-dimensional construct, directly influencing Einstein’s thinking.

Brane Theory and String Theory

Lisa Randall & Raman Sundrum – Proposed brane-world cosmologies, suggesting our universe may be a 4D brane in a higher-dimensional bulk.

Juan Maldacena – Developed the AdS/CFT correspondence, helping to bridge higher-dimensional spaces and holographic principles.

Edward Witten – Key contributor to string theory and M-theory, providing structure to the dimensional landscape of modern physics.

Big Rip Cosmology

Robert Caldwell – Co-authored the 2003 Big Rip paper, exploring how dark energy could drive a catastrophic tearing of spacetime.

Bounce Cosmology

Martin Bojowald & Abhay Ashtekar – Advanced loop quantum cosmology and the Big Bounce model, where the universe cyclically contracts and expands.

Paul Steinhardt – Co-developed the Ekpyrotic and Cyclic Universe models, where brane collisions replace Big Bang singularity.

General Relativity

Albert Einstein – Originator of general relativity, which fused space and time into a four-dimensional continuum and predicted black holes.

⭐ This post is a speculative cosmology inspired by general relativity, string theory, brane-world models, and non-Euclidean geometry. It’s not meant to describe current consensus physics—it’s meant to offer a new lens for thinking about time, black holes, and the structure of the universe. I write this from the perspective of someone who believes theory can also be poetry, and that the right metaphor can open new ways of seeing. Somewhere beneath the fold, something old is remembering itself.

* I am not an expert and if any mistakes are present, I take full responsibility 🖤 please take this post with a healthy grain of salt and have fun :)

[whats-in-a-sentence]

Each atom in a body-centred cubic lattice is at the centre of a cube and contacts eight neighbouring corner atoms (see figure 7.35). (...) The views of the iron crystal structure in figure 7.35 on the next page illustrates this arrangement.

"Chemistry" 2e - Blackman, A., Bottle, S., Schmid, S., Mocerino, M., Wille, U.

[tenth-sentence]

We can identify a crystal lattice for the larger ions, and then describe how the smaller ions pack within that lattice.

"Chemistry" 2e - Blackman, A., Bottle, S., Schmid, S., Mocerino, M., Wille, U.

[clottedscream]

you mentioned in the chat of a jello stream that you disliked how much fantasy magic crystals were shaped like Quartz. are there any specific crystal shapes you think are underutilized in fantasy?

I always like it when i see a nice beryl-or-andalusite-esque columnar crystal, or something botryoidal or acicular in a way that it makes little pom-pom shapes. And dogtooth (scalenohedral) crystals are a nice way to have that classic spiky shape without going quartz about it. But there's a whole entire world of weird and unique crystal shapes out there to base your fantasy crystals on, and I can't show off every possible shape that a crystal can come in with just one post. Pretty much every shape that a crystal can be is underutilized in comparison to the quartz shape.

the big thing that gets me about the "oops, all giant quartz clusters" method of designing fantasy crystals for your fictional world is that the shape a quartz crystal comes in, even though it's seen as like, the ubiquitous crystal shape that we think of all crystals as being, is actually very specific to quartzes!!!!

This is a figure from the mindat.org page on quartz. the top three shapes (a, b, and c) should be very familiar images to you! These are the most common shapes that crystals will be in fictional media. They're also three idealized crystal habits of QTZ - normal (prismatic), trigonal (prismatic), and pseudo-hexagonal (prismatic).

That's a lot of long words, but it all boils down to the describing geometry of a quartz crystal, and the geometry of its atomic structure.

QTZ is made of polymerized silicate, which, again, sounds fancy, but a polymer is just a long chain of many smaller molecules, and silicate is just the common term for the molecule SiO4. Silicate is a tetrahedron shaped molecule, meaning it's a pyramid with a square shaped base. A QTZ unit cell is made of a chain (a polymer) of silicate molecules linked together.

Here, we use unit cell to mean the simplest repeating building block in a larger pattern. like, say, if you build a giant lego cube out of all identical cube shaped legos, each cube shaped lego represents a unit cell of that giant cube. But QTZ unit cells aren't cube shaped. They're rhombohedrons- ie, 3d shapes where each face is a rhombus.

and if you stack a lot of rhombuses next to each other, they form a hexagon.

So when all the little unit cells of QTZ fit together, their ultimate shape is going to be hexagonal.

Quartz are not the only hexagonal mineral. Calcite famously also has a rhombus shaped unit cell. In mineralogy terms, this is called having trigonal symmetry- tri for three, as in the three rhombuses it takes to make a hexagon, or the three planes of symmetry a rhombus has.

But calcite doesn't form in in those tall prisms that terminate in pointy pyramid shapes made up of isosceles triangles or pentagons. It sometimes forms in dogtooth crystals, but dogtooth crystal faces are scalene, and they don't have that long prism body with a pyramid at the top like an endcap- they're spikes all the way to the base. Why is that? Why is quartz different from other trigonal minerals? why are its crystals weird like that?

Well, theres a lot of reasons, but one major one is that on top of being made of rhombohedrons, Quartz is ALSO made of helixes!

Thats right. Quartz has TWO fundamental patterns happening in its lattice. At the same time!

You can see by the overlayed yellow rhombohedrons in the figure above that each rhombus-y building block of QTZ fits together into a helix-shaped chain. QTZ forms in helixes because each of the basic silicate (SiO4) molecules in QTZ is sharing two of its oxygens with the other silicates its connected to. Because each block of the helix chain is made of a rhombus, when you stack those helix chains all next to each other to get a big quartz lattice, those chains make a hexagonal net. But helixes are also chiral, meaning they have a handedness to them- they can mirror each other and still be non-superimposable, like human hands.

To get the scope on why this chirality makes a difference to the structure of quartz, lets compare it to another mineral with trigonal symmetry, the aforementioned Calcite. If you could see atoms, this is what a chunk of calcite mineral would look like to you:

nice and simple, right? That's a very normal looking rhombus. everything slots together in a very straightforwardly rhombus-y way.

Now let's look at quartz.

ough.

When you combine these two features of QTZ geometry, the rhombus/trigonal symmetry and the helix shaped network of interlocking molecules, you get a pretty unique structure, which leads it to grow in very unique shapes when it gets bigger- hexagonal prisms with many interlocking chiral faces, terminating in those striking pyramid points composed of isosceles triangles.

That all sounds pretty cool right? it sounds like quartz is a really striking and unique and beautiful phenomenon of geometry, right? so why would I be annoyed? why would I be annoyed that a fantasy crystal has that unique shape? BECAUSE THAT SHAPE MEANS THE MINERAL IS QUARTZ.

Do you understand now? Out of all the possible permutations of different shapes a crystal can grow into, QUARTZ IS KIND OF THE ONLY FUCKING GUY THAT DOES THAT PARTICULAR SHAPE! nobody is out there doing it the way my guy quartz is doing it! So much so, in fact, THAT THIS IS THE ENTRY ON MINDAT FOR IDENTIFYING QUARTZ:

BY EYE.

QUARTZ IS ONE OF THE ONLY MINERALS YOU CAN ID BY EYE.

So, when you designed your mystical blue glowing crystal that has the power to harness a wizard's mana or whatever the fuck? And you picked that shape for it to be?

That means it's not enchantenite, or lunarite, or or whatever the fuck you want to call it. It's quartz. it's just quartz. your lack of creativity and unwillingness to do anything other than the most basic, recognizable shape for your fantasy crystal has all but guaranteed it. Maybe start worldbuilding what trace elements in that otherwise extremely fucking normal quartz you have there cause it to glow.

Because you picked the one shape that basically only quartz can be!

Congratulations!

enjoy your magical silicon dioxide, you piece of shit.

[literaryvein-reblogs]

More French Loans in Middle English

Loan Word - vocabulary borrowings

Borrow - to introduce a word (or some other linguistic feature) from one language or dialect into another

Leisure and the arts art, beauty, carol, chess, colour, conversation, courser, dalliance, dance, falcon, fool, harness, image, jollity, joust, juggler, kennel, lay, leisure, literature, lute, melody, minstrel, music, noun, painting, palfrey, paper, parchment, park, partridge, pavilion, pen, pheasant, poet, preface, prose, recreation, rein, retrieve, revel, rhyme, romance, sculpture, spaniel, stable, stallion, story, tabor, terrier, title, tournament, tragedy, trot, vellum, volume

Science and learning alkali, anatomy, arsenic, calendar, clause, copy, gender, geometry, gout, grammar, jaundice, leper, logic, medicine, metal, noun, ointment, pain, physician, plague, pleurisy, poison, pulse, sphere, square, stomach, study, sulphur, surgeon, treatise

The home basin, blanket, bucket, ceiling, cellar, chair, chamber, chandelier, chimney, closet, couch, counterpane, curtain, cushion, garret, joist, kennel, lamp, lantern, latch, lattice, pantry, parlour, pillar, porch, quilt, scullery, towel, tower, turret

General nouns action, adventure, affection, age, air, city, coast, comfort, country, courage, courtesy, cruelty, debt, deceit, dozen, envy, error, face, fault, flower, forest, grief, honour, hour, joy, labour, manner, marriage, mischief, mountain, noise, number, ocean, opinion, order, pair, people, person, piece, point, poverty, power, quality, rage, reason, river, scandal, season, sign, sound, spirit, substance, task, tavern, unity, vision

General adjectives active, amorous, blue, brown, calm, certain, clear, common, cruel, curious, eager, easy, final, foreign, gay, gentle, honest, horrible, large, mean, natural, nice, original, perfect, poor, precious, probable, real, rude, safe, scarce, scarlet, second, simple, single, solid, special, strange, sudden, sure, usual

General verbs advise, allow, arrange, carry, change, close, continue, cry, deceive, delay, enjoy, enter, form, grant, inform, join, marry, move, obey, pass, pay, please, prefer, prove, push, quit, receive, refuse, remember, reply, satisfy, save, serve, suppose, travel, trip, wait, waste

Turns of phrase by heart, come to a head, do homage, do justice to, have mercy on, hold one’s peace, make complaint, on the point of, take leave, take pity on

Part 1 ⚜ Source ⚜ More References: Middle English ⚜ Word Lists

[santoschristos]

“VOICEPRINTS: WHEN SPEECH REWIRES DNA” Every word you release is not just air — it’s a sound-laser etching geometry onto your double helix. Phonons hit hydrogen bonds, torsion angles shift, and suddenly a phrase becomes protein. THE SONIC GENOME LOOP 1.Phonon Keys – Vowels = compression waves, consonants = slicing waves. Together they twist chromatin like a DJ scratching vinyl. 2. Epigenetic Chords – Loving tones flood cells with acetyl tags (genes turn on), fear frequencies tighten methyl locks (genes go mute). 3. Quantum Echo – Your larynx broadcasts pico-Tesla fields; DNA coils pick them up like radio wire, rebroadcasting intent through blood. HOW TO SPEAK MIRACLES INTO MATTER • Hex-Code Intention – Convert goal to hex (HEAL = 0x48 0x45 0x41 0x4C) and recite once before bed; cortex stores hashes in dream RAM. • Heart-Tone Sync – Tap sternum 5×, hum 144 Hz, feel chest vibrate; heart’s EM torus modulates vocal carrier wave. • Spiral Breath – Inhale through nose 5, hold 2, exhale through pursed lips 8 while whispering the outcome as if done. • Dawn Deployment – First conversation of the day, slip your prime word into sentence #3; lattice hears, probability tilts. SIDE EFFECTS (48 h) • Random compliments on your “energy.” • Messages arrive carrying exact phrases you embedded. • Synchronicities stack: green lights, song cues, inbox yeses. Stop thinking of talk as noise. It’s source-code for biology. Write your body’s next update with syllables that sparkle. Shanaka Anslem Perera @shanaka86 “The quality of your life does not depend on what you have or what you do, but rather your state of being.” Image: Magic, Power, Language, Symbol Speak Your Truth Mahaboka

[algebraic-dualist]

Universal Algebraic Geometry, Part 1

In this post I recall a number of basic ideas from universal algebra in order to start setting up a framework for studying the algebraic geometry of arbitrary algebraic objects.

For our purposes, an algebra is just a set equipped with some operations. We assume that the set of operations has been fixed. Given an algebra A alongside an operation ♡ we write ♡ᴬ for the definition of that operation on the algebra A, or just ♡ when ambiguity does not result. Our notion of a homomorphism is as follows:

Definition: A homomorphism f from an algebra A to an algebra B is a function f : A -> B such that for each n-ary operation ♡ alongside elements x₁, ..., xₙ ∈ A it follows that f(♡ᴬ(x₁, ..., xₙ)) = ♡ᴮ(f(x₁), ..., f(xₙ))

Kernels and Congruences

Definition: A congruence θ on an algebra A is an equivalence relation such that for any (x₁, y₁) ∈ θ, ..., (xₙ, yₙ) ∈ θ and n-ary operation ♡ it follows that (♡ᴬ(x₁, ..., xₙ), ♡ᴬ(y₁, ..., yₙ)) ∈ θ. We write Cgr(A) for the set of congruences on A.

Example: The maximal congruence ∇ on A is just A × A.

Example: The minimal congruence ∆ on A is just { (x, x) | x ∈ A }.

Example: Any arbitrary intersection of congruences on A is a congruence on A. In particular, we can form the intersection of all congruences on A containing any arbitrary subset R of A × A in order to obtain the minimum congruence containing R. In particular, choosing R to be the union of two congruences, we get a meet of congruences.

In the case of a group, it follows that (x, y) ∈ θ if and only if (x - y, 0) ∈ θ hence a congruence is completely determined by the set { x | (x, 0) ∈ θ }. In other words, by a normal subgroup. So congruences are just the data of a normal subgroup. Similar remarks apply to ideals of rings.

Kernels of homomorphisms are a particularly important source of congruences on algebras:

Definition: The kernel of a homomorphism f : A -> B is denoted by ker(f) and defined by setting ker(f) := { (x, y) | f(x) = f(y) }.

Lemma: Given a homomorphism f : A -> B it follows that ker(f) is a congruence on A. Proof:

Given x ∈ A it follows that f(x) = f(x) hence (x, x) ∈ ker(f).

Given x, y ∈ A such that (x, y) ∈ ker(f) it follows that f(x) = f(y), which is to say f(y) = f(x) hence (y, x) ∈ ker(f).

Given x, y, z ∈ A it such that (x, y) ∈ ker(f) and (y, z) ∈ ker(f) it follows that f(x) = f(y) = f(z) hence (x, z) ∈ ker(f).

Given (x₁, y₁) ∈ ker(f), ..., (xₙ, yₙ) ∈ ker(f) and n-ary operation ♡ it follows that f(xₖ) = f(yₖ) for each k, thus by the homomorphism property f(♡ᴬ(x₁, ..., xₙ)) = ♡ᴮ(f(x₁), ..., f(xₙ)) = ♡ᴮ(f(y₁), ..., f(yₙ)) = f(♡ᴬ(y₁, ..., yₙ)) showing that (♡ᴬ(x₁, ..., xₙ),♡ᴬ(y₁, ..., yₙ)) ∈ ker(f).

QED.

Theorem: Given a homomorphism f : A -> B alongside a congruence θ on B it follows that the preimage f*(θ) := { (x, y) ∈ A | (f(x), f(y)) ∈ θ } is a congruence on A. Proof: 1. Given x ∈ A, as θ is reflexive it follows that (f(x), f(x)) ∈ θ hence x ∈ f*(θ). 2. Given (x, y) ∈ f*(θ) it follows that (f(x), f(y)) ∈ θ. As θ is symmetric it follows that (f(y),f(x)) ∈ θ, thus (y, x) ∈ f*(θ). 3. Given (x,y) ∈ f*(θ) and (y,z) ∈ f*(θ) it follows that (f(x), f(y)) ∈ θ and (f(y), f(z)) ∈ θ. As θ is transitive it follows that (f(x), f(z)) ∈ θ hence (x, z) ∈ f*(θ). 4. Given (x₁, y₁) ∈ f*(θ), ..., (xₙ, yₙ) ∈ f*(θ) alongside an n-ary operator ♡ it follows that (f(x₁), f(y₁)) ∈ θ, ..., (f(xₙ), f(yₙ)) ∈ θ, hence as θ is a congruence (♡(f(x₁), ..., f(xₙ)), ♡(f(y₁), ..., f(yₙ))) ∈ θ. Being that f is a homomorphism, this means (f(♡(x₁, ..., xₙ)), f(♡(y₁, ..., yₙ))) ∈ θ showing that (♡(x₁, ..., xₙ), ♡(y₁, ..., yₙ)) ∈ f*(θ). QED.

Therefore, every homomorphism f : A -> B induces a map f* : Cgr(B) -> Cgr(A). In fact, congruences organise themselves into a lattice, and this map f* is a homomorphism of lattices. But for the moment, we will not worry about that.

Composite Kernel Lemma: Given homomorphisms f : A -> B and g : B -> C it follows that

ker(g ∘ f) = f*(ker(g)).

Proof:

Observe that (x, y) ∈ ker(g ∘ f) if and only if g(f(x)) = g(f(y)), which is the case if and only if (f(x), f(y)) ∈ ker(g), which is the case if and only if (x, y) ∈ f*(ker(g)).

QED. Corollary: ker(f) = f*(∆). Proof: ker(id ∘ f) = f*(ker(id)) = f*(∆). QED.

Quotients of Algebras

Given a congruence θ on an algebra A it is the possible to construct the quotient algebra A/θ whose elements are equivalence classes of elements of A under the congruence θ. Sending every element to its equivalence class under θ gives a homomorphism π : A -> A/θ which we refer to as the canonical projection of θ.

Theorem: Given a congruence θ on an algebra A it follows for every homomorphism f : A -> B with θ ⊆ ker(f) that there exists a unique injective morphism g : A/θ -> B such that f = g ∘ π.

Proof: Observe that π(x) = π(y) iff (x, y) ∈ θ, which as θ ⊆ ker(f) implies f(x) = f(y). Therefore, it is well-defined to construct g by setting g(π(x)) = f(x). One can check that this works. QED.

Homomorphism Theorem: Given a surjective homomorphism f : A -> B it follows that there exists an isomorphic g : A/ker(f) -> B such that g ∘ π = f.

Proof:

Applying the prior Theorem there exists a unique injective morphism g : A/ker(f) -> B such that g ∘ π = f. Observe that as f is surjective, it then follows that g is surjective.

QED.

Hereditary Classes of Algebras

Recall that the prime ideals 𝔭 of a ring R are exactly those ideals 𝔭 such that R/𝔭 is an integral domain. Notice that being an integral domain amounts to satisfying a universal axiom, namely xy = 0 implies x = 0 or y = 0. Hence, integral domains are closed under moving to subalgebras, and the property of being an integral domain is preserved by isomorphisms.

Definition: A class K of algebras is hereditary iff it is closed under subalgebras and isomorphisms. Given a universal class K we say that a congruence θ on an algebra A is a K-congruence iff the quotient algebra A/θ belongs to the class K.

Theorem: Given a hereditary class K alongside a homomorphism f : A -> B it follows for every K-congruence θ on B that f*(θ) is a K-congruence on A.

Proof:

Consider a homomorphism f : A -> B alongside a K-congruence θ on B. It follows that f*(θ) is a congruence on A, so it suffices to demonstrate that it is a K-congruence. Recalling our canonical projection π : B -> B/θ and our Composite Kernel Lemma, it follows that ker(π ∘ f) = f*(ker(π)) = f*(θ). Therefore, the morphism π ∘ f : A -> B/θ factorises through a unique injective morphism g : A/f*(θ) -> B/θ. This means that A/f*(θ) is isomorphic to a subalgebra of B/θ. As θ is a K-congruence it follows that B/θ belongs to K, which as K is a hereditary class then implies that A/f*(θ) belongs to K, so that f*(θ) is then a K-congruence.

QED.

Corollary: Preimages of prime ideals are prime ideals.

[tenderwatches]

summary: Viktor and Jayce get a little too close in the lab + a look at how viktor learnt the rules of surviving in Piltover as an Undercity transplant cw: this chapter contains ableist language (canon, self-referential) and descriptions of medical suturing

When Viktor first became aware that he’d die early, he’d been angry.

It hadn’t been pretty. He’d yelled at his mother, at his father, had thrown things—his cane, books, half-assembled inventions. He’d cried and screamed until he’d worked himself up so badly that his father had needed to sprint to a breathing station with Viktor on his back in hopes that the cleaner air would dampen his desperate wheezing.

After that, he’d been listless for days, lying in bed, trying to conjure up the motivation to work for anything when his time would be so short, so inconsequential.

And then he remembered Rio.

The waverider was a huge creature he visited where a strange man in a strange place beyond the ravine kept her. She was like a salamander glistening in shades of blush and blossom, with big eyes full of curiosity and a tongue that craved sweet nectar. Such a simple creature, but he still thought of her, even years after he’d last seen her. He still thought of her and of the man who was so determined to keep her alive that he had not cared if she lived.

He thought of infants, cold in their cradles, their lives snuffed out, breaths robbed by the Gray. He thought of children wasting away, disfigured by the slicks of toxic chemicals oozing from chemtech seams deep in crevasses, and how he, at least, knew sunlight.

Since then, Viktor has done his best to ensure that every moment of his short life contributes to something greater than himself. The people whose lives he’s saved in the Undercity will go on to have families; they’ll impart their knowledge upon others who will do the same, who will do the same, who will do the same.

Life, like an object in motion, stays in motion.

Energy can neither be created nor destroyed—it can only be transformed.

Viktor hopes that after his death, the energy that was once heat in his body will permeate into the ether, atoms ricocheting into the endless universe.

Until then, he’s resolved to stay in motion.

This determination presses his lips into a thin line of concentration as he makes minute adjustments to the dials on the microscope before him. Crystalline structures resolve into sharp focus, forming wild geometries that defy every principle of natural formation. Unlike genuine hex crystals with their orderly lattices, the synthetics’ birefringent patterns are irregular, and the arcane’s response is unlike anything he’s ever seen.

Viktor pauses to scribble a few words alongside one of Jayce’s diagrams, adding to the existing maze of their notations. They’ve managed to offset interference caused by the unexpected new compounds identified in their lab tests, but there is still residual output to deal with. He takes this moment to rest his forehead in his hand, momentarily closing his eyes. The urge to surrender to sleep swells slowly, like a building tidal wave, and he forces his eyes open before it can break over him.

One of Jayce’s hexscopes (an adaptation of one of his early designs) sits open on the table, tracking arcane energy as it moves through the crystal matrix. He stares hard at the pen attached to the end of its thin metal arm, scratching softly on continuously rolling paper as it records the waveforms. The resulting bonds are irregular with equally unpredictable chemical reactions—

Chemical reactions! Viktor straightens up so quickly that the momentum almost carries him over backwards. Head reeling, he stumbles to steady himself against the desk, pulling the diagrams in for a better look before jamming the microscope against his eye again, squinting hard at what he sees under the lens. These erratic bonds must be the cause of the arcane’s volatile reactions and inconsistencies. They’re brittle instead of strong, releasing energy in unexpected spurts. They’re illogical, full of contradictions. Only chemtech can force such incompatible combinations to hold.

Simultaneous thoughts fire off in all directions—what this means for the outputs they’re attempting to handle (harness? Eliminate? Neutralise?), the tenuous nature of these bonds, the undersized reactions sustained by the crystals—and the nagging feeling he’s seen this all somewhere before. Somewhere in Zaun, near the seams, where, in his youth, anger at the injustice of his life had gotten the better of him. Somewhere he'd nearly gotten buried in collapsing pipes, flashing fuchsia and green in the darkness of the sump.

He drops away from the microscope and back into his chair. Perhaps he should consider bringing some of this work back to the lab Heimerdinger had set aside for him. It’s closer to his Academy-issued apartment than Jayce’s lab is, and with the constant travel across the city, he often finds himself exhausted before he’s even really gotten started. Today, just like many other days, his leg aches as if he’s been standing for the entire morning, though it’s only been a couple of hours since he arrived. The considerations of the crystals, his small inconveniences, the way they all still stagger him, make the walls begin to feel oppressive. The clean lines and polished brass are a far cry from the corrugated metal and improvisation he was used to in the Undercity, and yet—he’s now facing the same kind of problem. These synthetic crystals with their arcane violations bear toxicity here, whilst below, poison is a by-product of unholy greed.

In both places, they stand to lose so much, and yet the eyes of the elite are perpetually closed.

Viktor’s teeth grind as he grips the edge of the workbench to pull himself up again, ignoring how his muscles protest. He begins recalibrating the containment field to account for an array of chemical reactions, instead of only the ones they’d adjusted for earlier in the week. If he can just isolate the unstable compounds, maybe apply some of the principles he’d developed during his academy years, they can counteract or capture the arcane fluctuations.

He’s so deep in focus that he almost jumps when Jayce walks in, chatting before he’s even crossed the threshold. “I thought you might want to see the latest stability readings from—” He breaks off, and Viktor knows his eyes are fixed on the modified containment field setup. “What are you doing?”

“Testing a theory.” Viktor doesn’t look up from the controls. The crystal’s glow intensifies, casting flickering shadows across his hands. “The synthetic crystals are made with chemtech. The instability isn’t a flaw—it’s a signature.”

“Testing a wh—wait, chemtech?” Jayce’s footsteps quicken across the floor. He unceremoniously drops his papers onto the desk, knocking the pen of the hexscope out of alignment. It continues to dutifully work through arching waves, up and down. “Hold on, you can’t just—we don’t even have protocols for working with—”

“We don’t have time for protocols,” he hisses, sharper than he intends. He forces patience into his voice. “It’s like Councillor Medarda told you—every day, Clan Ferros grows more restless—”

“Viktor, wait—”

The crystal flares with brilliant lances of blue-white light, shattering into shards that streak across the lab, acid green and electric purple tails in their wake. Viktor recoils from it and finds himself crashing first into the chair behind him, then the ground.

His breath leaves his lungs without being replaced by another—it’s a second too long before he can gasp again, sucking in air that smells of Jayce’s aftershave. Sandalwood mingles with the smell of sulphur and iron. Stars shrink and grow in his eyes.

“Are you—” When Jayce speaks, Viktor slowly becomes aware that he’s caged by a pair of smooth, sturdy forearms. His former partner is propped above him, but not so much that their bodies aren’t pressed flush together. Heat grows between them. Jayce’s chest heaves as his own gives, and for a moment, Viktor can’t speak.

A gentle furrow forms between Jayce's brows as he quickly pushes himself up onto one palm, the other coming up to cup Viktor's cheek. The motion is gentle and unthinking, fraught with the effortless care these kinds of gestures bore in their past. His eyes search Viktor's face with worried intensity, thumb brushing along the angular line of his cheekbone and coming away bright red and wet. “Hey, V—” he soothes, voice soft with an intimacy that makes Viktor's chest tight. “V-Viktor—hey.” The feeling dissipates.

Viktor pushes Jayce’s touch away and tries to sit up. He slides his hand back to support himself as he does, successfully forcing Jayce back on to his heels. Viktor finds the resulting breadth of air between them too cool on his skin, and wheezing feels like a flurry of knives in his chest. “I’m fine, Jayce,” he dismisses, muffling an accompanying cough in the crook of his elbow. He winces at the taste of copper in the back of his throat, hot embarrassment coursing through him. He can’t meet Jayce’s eyes; they are too bright with concern, honest anxiety spilling forth, unguarded. “You did not have to—” But both the words and his irritation die in his throat as dark droplets begin to dot the tiles at Jayce’s feet. “You’re bleeding.”

“Oh.” Jayce twists slightly to look over his shoulder, but immediately relaxes his posture when the movement elicits a wince. “It’s nothing. Besides, you are, too,” he points out, vaguely indicating Viktor’s cheek.

“A scratch.” Viktor feels nauseous. Not only did Jayce need to protect him like a helpless child, but he’d gotten himself hurt in the process. He leans forward, breath escaping him in a frustrated huff. “You’re bleeding on the floor. Let me see.”

“Viktor, really, it’s—”

“Take off your shirt.”

Jayce’s amber eyes turn into burning discs, his dark brows lost in his messy hair. Viktor feels the back of his neck flush with heat. “Your back, Jayce. Let me see,” he repeats, mortification sharpening the edges of his demand.

Viktor tries to ignore the inherent eroticism of demanding that Jayce turn around and strip—only to immediately fail when Jayce simply does it. By the time the two of them settle again (the lab’s robust first aid kit to one side of Viktor, Jayce sitting cross-legged in front), Viktor is dizzy. He wishes he could say it’s purely due to the sheer amount of bronze skin on display, but the headache blooming up from the base of his skull tells him otherwise. He concentrates on applying a local anaesthetic to the scattering of gashes across Jayce’s broad shoulders, then dabbing each with an antiseptic. “Well, the good news is you’ll live,” he jests, managing to thread a thin, curved surgical needle. His movements are slow but steady, and when he leans in closer to begin his work, the room rocks by only a small margin.

“Thanks, doc, what a relief.” Jayce turns his laugh into a soft snort, presumably so as to not disrupt Viktor’s stitching. Though he hasn’t needed to exercise this skill in months, the repetitive motions return to him with ease. They sit in stillness a while longer before Jayce hesitantly raises the question, “What… were you saying about chemtech?”

In spite of himself, Viktor smiles. What he wouldn’t give for the world to have the kind of insatiable curiosity that Jayce Talis has.

He walks his former partner through the process of his discovery, naming the impossibilities, the idiosyncrasies, and the ways in which he suspects the arcane clashes with the chemical compounds. Jayce is just as intrigued as Viktor, and Viktor can’t help but think that, had Jayce been in his position, they might have ended up in this same situation despite the other man’s usual adherence to safety precautions. Words of science, math, and discovery pass between them with the easiness of butterflies on a breeze, punctuated by an occasional excited exclamation from Jayce.

His progress on Jayce’s back is clean and methodical—habits formed from years of treating injuries in the Undercity, where wounds proved you couldn't stay out of trouble and seeking proper care marked dependency. He uses a pair of forceps to guide the needle through each wound, stopping only to tie off each suture as he moves from one cut to the next.

“When did you learn to do this?” Jayce has never been good at silences.

“Long ago.” Viktor keeps his eyes fixed on his work, feeling perverse as he notices the warmth radiating from Jayce’s skin, even through the sterile gloves he’s donned. “I have always fixed things. Mending clothes or skin, it makes little difference to me.”

His hands have moved now, lower down, from the broad muscle of the trapezius. He rests his fingers there for a second too long, and the name of the muscle floats through his mind, 'latissimus dorsi', as if the words are a subconscious effort to pull him from other thoughts.

“I had to learn some of this too, actually. In the forge—hot metal doesn't always go where you want it to,” Jayce offers, and Viktor’s hands still, his thoughts returning to the present.

The differences in their circumstances are not lost on him, but he recognises Jayce’s attempt to… relate to him. “I suppose we both learnt through trial and error,” he acknowledges.

“Not that—not that it’s, uh, the same.”

Viktor hears the uncertainty in Jayce’s hurried addition, as if he’s waiting for a sign from Viktor to indicate he’s irritated that Jayce has drawn the comparison. “You can relax, Jayce.”

Viktor pulls the gloves from his hands and sits back to survey his handiwork. He’s stitched four lacerations in total, covering each with neat squares of gauze taped down over Jayce’s tanned skin. The damage, thankfully, wasn’t worse than any of the other countless accidents they’ve had in the lab, but Viktor still feels that curl of shame at being impatient enough to have caused this one.

Silence expands to fill the gulf between their differences—Viktor’s skills hard-won through necessity, Jayce’s forged with the security of his family and promises of a bright future. And yet, an uncanny symmetry has brought them to this point, just as it had years ago; one extraordinary moment in which their paths converged.

“Why would you risk this?”

Back then, he’d told Jayce that he hadn’t aspired to be an assistant for the rest of his life—and that was true. But beyond that, he’d known he was running out of options.

Every action, every movement, all the things he’s ever contributed, has an impact, however imperceptible in the long line of the universe. But it’s not enough for him to simply have been; he wants to be remembered.

Though energy can neither be created nor destroyed, human legacies are far more fragile things.

𐡸.:𐫱:.𐡷

Summertime, 978 AN - fifteen years ago

Nothing tasted as bitter as cruel irony, Viktor thought, as he made his way up what had to be the seventh staircase between him and the first stage on which he’d have to parade himself like a show dog. Progress Day in Piltover arrived with fanfare, as always, and the usual thrum of city life had become more of a coursing roar.

From Glasswell Street to Sidereal Avenue and Incognia Plaza, crowds gathered around vendor carts and at the colourful merchant tents, blissfully caught up in the spirit of innovation and promise.

Not one seemed to remember that, centuries ago, this day had not been marked by celebration but by terror and half a city being swallowed by the sea.

In the four years since his arrival at the University of Piltover, Viktor had marked each Progress Day by lighting a candle in remembrance of those Zaun had lost in the disaster. Earthquakes, resulting from the blasts detonated to clear the way for the Sun Gates, had thrashed the streets of the Undercity, sacrificing thousands of lives to the ocean—all in the name of progress.

Now, here he was, prepared to submit himself to the judgement of those who had so greatly benefitted from the influx of trade the Sun Gates had ushered in. He had only two destinations in mind, but the journey to the merchant families’ tents was already enough to send pain lancing up his leg. Maybe it was his penance walking over those watery graves to attend their school, study their sciences, and pretend like he was one of them.

He felt that he was doing a rather shoddy job of it, by the way the artificers peered at him with narrowed eyes that flicked between him and his papers. This overt display of suspicion made him curse Professor Heimerdinger for forcing him into this lavish ordeal. Auditioning had never been in Viktor’s plans—he knew better than to fool himself into thinking he could join the ranks of Piltover’s apprenta.

Rule number one: They will not make space for you.

He could build bridges upon bridges over the work his classmates created, but the city’s artificers, ruled by the wealthiest of the merchant class, would sooner retrofit their workshops with last year’s scrap metal than take on a cripple from the Undercity, even when the dean of the academy and head of the council himself had singled him out.

Graduation loomed ever closer, and despite his time at the academy, the future felt uncertain. Piltover’s clean air and bright sunlight had undoubtedly improved his health (incredible what being able to breathe did for a person), and the prospect of returning to the Undercity daunted him. He needed an apprenticeship probably more than anyone else stood waiting in the chamber, and yet, he was certain that he was the least likely to receive one, no matter how sound his work was.

“Name?” One of the artificers asked as she handed back the paper that clearly bore his name. He tightened his grip on his invention in an effort to hold his tongue.

“Viktor.”

“Full name?”

“It’s… just Viktor.”

She treated him with the kind of disdain that only someone with a meagre amount of power could manage. He hated her for it, and then hated that he did. It was too petty to warrant such a response from him, but his entire body was sore now. He’d pushed himself through the uneven cobblestone streets faster than he should have dared. He’d even risen with the sun, well before he’d needed to. He wanted to give them as few reasons to dismiss him as he could manage, thinking his dedication to punctuality might also communicate his regard for their time and win some small amount of their favour.

From the placid way the artificers looked at him, he could see that was not the case.

Of course, he’d known this and had even explained such to Heimerdinger when the dean had urged him to take on the auditions. How difficult it would be for him to simply make the physical journey in a process that was designed to showcase resilience and determination; how his accent would immediately mark him as ‘other’, and how the inventions he was proudest of were things that would not sparkle and flash the way Piltover expected. His progress was for the Undercity, and thus, it might as well have been invisible.

Already, their attention drifted, and other hopefuls surrounding the tent seemed to bear down on him. He grimaced as he set up his contraption of pipes and dials that looked out of place within the sleek lines of the tent. It wasn’t until his machine began hissing shrilly and emitting puffs of Gray that the artificers paid him any mind. One of them started shrieking, making it very difficult to explain that he’d also released an aerosolised alkali to neutralise the toxicity—the whole point of the showy demonstration.

Rule number two: their grace is precarious.

Whilst he’d never been foolish enough to think that the artificers might like what he brought to the auditions, he’d not been expecting their fury. With stomach-piercing fear, he realised that they, in all their self-aggrandising glory, seemed to think that his audition was an assassination attempt of the mercantile family. The absurdity nearly made him long for simpler days, when people merely saw his mistakes as proof of unworthiness, and his greatest lament was how they judged his errors more harshly than his peers' mere learning experiences.

He’d packed his machine in a hurry and practically fled the tent, almost tripping himself like he’d not done since he was a child in his haste to slip into the crowd.

Rule number three: They will lie to you.

By the time the day ended, he’d attended only one more audition, though he was hardly sure that it counted. He hadn’t spent very much time at the Holloran tent, but the experience still clung to him like a stubborn mood, even as he sat in the safety of his favourite haunt in Piltover. The mechanical oasis overlooked the promenade level of the Undercity, waters running through the ravine below, where he’d played as a child. He’d always appreciated the serenity of this place, finding even in his youth that its quiet tranquillity suited him.

“Viktor, my boy,” called a reedy voice from behind him, and he lifted a hand off his cane in greeting without turning to look at Professor Heimerdinger. “How did your ventures go today?”

“I don’t believe it really ‘went,’” he responded wryly, easing himself into a seated position in the arch of the open-air window, legs relaxing over the ledge. “Can you say it ‘went’ if one family thought I was attempting a murder, and the other refused me at the door?” Heimerdinger’s poro scurried over his lap and around his back, which he found both ridiculous and… cute. It made his bitter remark come out with a slightly amused lilt, even if there wasn’t much to find amusing in being turned away, only to almost be knocked over by the next hopeful student when the Holloran family admitted them mere moments after.

The professor gave a soft hum, a gloved hand at his chin in the perfect pose of refined thought. “What will you do?”

Viktor rolled his shoulders in a noncommittal shrug. “I still have some time. I will need to finish the year, of course, and then… Well, then, probably the, ah, how do they say? ‘Crunch time’? Comes?”

Heimerdinger’s moustache twitched in a smile that didn’t quite meet his eyes, even though Viktor thought that his use of the colloquialism had been rather apt. When the professor spoke next, his words were soft and cautious, as if he thought that Viktor might snap. “Why don’t you consider being my assistant, lad? You could stay on at the academy, and though I’m sure you’d rather be doing something more ambitious, you’d have time to pursue your own projects.”

Silence hung in the air between them for a moment. Viktor tried to read the expression in Heimerdinger’s eyes but only saw a soft sorrow there. “I appreciate your offer, Professor,” he started, the words tasting of defeat even before he’d spoken of any decision.

Heimerdinger seemed to sense Viktor’s aversion and interjected before he could continue. “So you’re aware, Viktor, this isn’t mere charity.” The professor turned inwards, eyes downcast, a slight droop to his large ears. “I was… perhaps hasty,” he admitted, still looking at the cement floor, “in urging you to audition.”

Viktor had never known Professor Heimerdinger to be prideful, but the dean’s guilty posture struck him, even so. His kindness still burnt; Viktor’s stubborn independence made him reactive to the idea of being handed anything out of pity, particularly given the assumptions of other students who already believed his mere presence was an excess of anything he had any right to. “Thank you, Professor.” He found that he meant it. Heimerdinger had always believed in his potential, even when doing so set him at odds with the rest of the faculty. “Perhaps… give me some time to think it over,” he relented, looking back out at the city below. The streets still bustled with the activity of Progress Day, even as the sun began to cast warm, dusky shadows amidst the revelry.

“Take the time you need, my boy. The offer stands.” With that, the professor retreated at a quick trot, his ever-present poro shuffling along behind him. Viktor sat in the wake of their departure, contemplating the glint of mechanical contraptions dotting the landscape (so far as he could tell, they were only constructed as decor for the day, which was an awful waste, considering what you could buy in the Undercity after selling parts of just one). Perhaps it had been a blessing that he’d not managed a successful audition. Being the assistant to the academy's dean would mean he would have access to lab spaces and materials that most others would not, including unusual things that would need to be assessed for danger.

That could be interesting.

𓊈 first chapter | previous chapter | next chapter on AO3 𓊉

an: this chapter was SO fun to write - definitely one of the ones we were most looking forward to when we were posting on AO3!! i'm so bad at these tumblr updates im going to try to get a bunch of them scheduled at once and see what happens haha anyways tho we just posted chapter 23 yesterday on AO3! 🙌🏽 fic come so far 😭

[adastra-sf]

Nano-3D printed material as light as styrofoam, 5× stronger than titanium

image of the full nanolattice geometry (left), and an 18.75-million cell nanolattice floating on a soap bubble (right)

Researchers at the University of Toronto’s Faculty of Applied Science & Engineering have created nano-architected materials stronger than any solid material, but lighter than a feather.

In a new paper published in Advanced Materials, a team describes how they made dream nanomaterials with properties that offer a typically conflicting combination of exceptional strength, light weight, and customizability. The approach could be applied to a wide range of applications - automotive, architecture, aerospace, and much more.

Manufacturing technology like this will revolutionize the world - imagine being able to print out the strongest, lightest gear someone might need, on a moment's notice, to tackle any task [your character] might face. All they'd need is a 3D nano-printer (in this case, a two-photon polymerization type) in their workshop and the right printing materials (in this case, a form of carbon).

Nano-architected materials are made of tiny building blocks or repeating units measuring a few hundred nanometers in size - a human hair is more than 100× thicker than the lattice structures pictured above. These building blocks (in this case made of carbon) are arranged in complex 3D structures called nanolattices.   

This is also the technology we've needed in order to begin building space elevators to get past expensive, dangerous rocket tech.

[faultfalha]

In a move that is sure to revolutionize the printing industry, Metafold, a leading geometric 3D-printing specialist, has announced that it has landed $1.78M in funding. This infusion of cash will allow the company to expand its operations and bring its innovative printing technology to a wider audience. Metafold's unique printing technique, which relies on geometric folding to create three-dimensional objects, has generated a great deal of interest in the printing community. And with good reason: the results are simply stunning. Thanks to this new round of funding, Metafold is poised to take the printing world by storm. Be sure to keep an eye out for its products in the near future!

[stayycalm]

i have a Request! could you write a little short about Hyunjin and Fem Reader? maybe something where Reader is in paris and taking picture of the beautiful scenery and ends up taking a picture of Hyunjin? but like they dont notice till they are back at their hotel? and then they are admiring the beautiful stranger and they cant sleep so they go out on their balcony to look at the beautiful lights but little do they know a wandering Hyunjin is outside looking for inspiration for drawing/paintng and when he sees a mysterious figure on a balcony hes enthralled by their silhouette? he hurries back to his room and begins to sketch right away and he wonders who the beauty was, and then they end up meeting? sorry this is long but i really want to know how you'd write this!!

ok, first of all, I want to deeply apologize, anon, because I'm just now finding this ask??? it's been in my inbox for a year?? so please accept my deepest apology and enjoy this! 😭

Drawn to You by stayycalm

The streets of Paris hum with life as I sling my trusty Canon Rebel T7 over my shoulder, ready to conquer my ambitious agenda. Two weeks here, and yet, the city’s allure hasn’t dulled a bit. Every street corner, every café, every wrought-iron balcony layered with Gothic spires or curving Art Nouveau façades feels like a treasure to an aspiring architect like me.

On this second-to-last day, I’m determined to make the most of it. The itinerary, planned by a finance teacher who wouldn’t know a flying buttress from a baguette, has been criminally devoid of exploration. I mutter under my breath, “How can anyone come here and not explore?”

I pause at Pont Alexandre III, lifting my camera to capture the graceful curve of a bridge flanked by its gilded statues, glowing in the soft morning light. Click. I take another, angling for the perfect shot of a bride crossing with her billowing white train, the Eiffel Tower rising in the distance.

As the day unfolds, I let my instincts guide me. From the serene paths of the Tuileries Gardens to ornate windows with their curved frames, I capture it all. Then, just as the sun begins to dip low, its golden light casting long shadows across the city, I find myself facing the Eiffel Tower. Its iron lattice glows with a warmth that takes my breath away. I can’t help but do a little happy dance as I check the shot in my viewfinder, triumphant.

A text buzzes from my phone, snapping me out of my euphoria. It’s my roommate, letting me know she’s heading back to the hotel. Sighing, I tuck my phone away and begin the trek back.

Later, I sit at the desk in our room, my hair wrapped in a towel from a much-needed shower. My laptop hums softly as I transfer the day’s photos. One by one, they fill the screen, and my excitement grows with every frame. I start editing, erasing stray tourists and clutter from the pristine geometry of my shots, until I reach the final picture.

My fingers hover over the trackpad as I zoom in on a figure in the background. He’s leaning casually against the bridge railing near the Eiffel Tower, his silhouette partially lit by the fading sunlight. His dark hair falls just past his shoulders, and his profile—what little I can see—is striking.

“Wow…” The word escapes me, barely a whisper.

“What are you oooh-ing over?” My roommate’s voice startles me. She emerges from the steamy bathroom, her towel-turbaned head tilting as she raises an eyebrow at me.

I jolt, fumbling to close the photo. “Oh, you know… buildings.”

She rolls her eyes dramatically. “Only you could get off to a building.”

I snort, but the heat rises to my cheeks. Turning back to the laptop, I save my edits and close it, needing an escape. “I’m stepping out for a sec,” I announce, grabbing a cardigan and slipping out to the balcony.

The air is cool, and the city stretches before me, a sea of glittering lights. With the sun gone, Paris seems transformed into a constellation fallen to earth, each light twinkling like a star. I lean against the stone railing, letting the sight fill me.

It’s moments like these that make Paris feel like a dream—a city so alive, yet timeless. A city of romance, indeed.

Hyunjin's POV

The sharp click of my shoes against the marble echoed through the Versace store as I stepped out, the heavy glass doors swinging open. A wave of flashing lights greeted me, relentless and blinding, a cacophony of voices calling out my name—professional photographers and STAY alike. It was always surreal, this paradoxical familiarity. I didn’t know a single face in that sea of people, yet they all knew me, or at least the version of me they thought they knew.

My lips curled into a smile, practiced but genuine in its intent. The sunglasses resting on the bridge of my nose shielded my eyes, but they couldn’t hide my gratitude. I raised a hand, waving to the crowd before blowing a kiss into the air. The gesture elicited a cheer, a ripple of excitement that followed me until I ducked into the sleek black car waiting at the curb.

Inside the car, the energy of the crowd faded, replaced by the hum of the engine and the muted bustle of Paris beyond the tinted windows. When we pulled up to the hotel, I took a moment to shed the Versace image. Gone were the statement pieces; in their place, a plain hoodie, jeans, and sneakers.

Tonight, I wasn’t Hyunjin, the idol, the "Versace prince." Tonight, I was just Hyunjin—an artist looking for inspiration in the heart of Paris.

The city’s streets welcomed me with their usual charm: cobblestone paths, golden streetlights, and the murmur of life flowing seamlessly between its residents and visitors. It should’ve been easy—a city as alive as Paris practically begged to be captured in art. Yet, as I wandered through alleyways, past street performers and cafés spilling over with laughter, nothing reached out to me. The fire I sought remained elusive, a spark I couldn’t ignite.

The Eiffel Tower loomed in the distance, its iron frame bathed in the amber glow of the setting sun. I found myself on a bridge, leaning against the railing as I stared out over the Seine. The water reflected the dying light in shimmering streaks, and the Tower cast a long shadow over the city.

“Where are you?” I muttered under my breath, the question aimed at no one in particular. A sigh escaped me as I pushed off the railing, shaking my head. Inspiration had never been this difficult to find before.

By the time the sun had dipped completely below the horizon, Paris transformed into a city of light. Neon signs buzzed, music drifted from open windows, and the streets teemed with energy. But even in the midst of all this vibrancy, I felt disconnected. My steps slowed as I glanced around one last time, taking in the tourist shops and the occasional artist sketching caricatures on the sidewalk. Still, nothing.

Frustration prickled at my skin. I tilted my head back, closed my eyes, and ran a hand through my hair, exhaling sharply. When I opened my eyes again, I froze mid-step.

She stood on a balcony a few stories above me, her silhouette framed by the soft glow of the room behind her. Her head was tilted upward, as if catching the glimmer of the city’s lights on her skin. Her hair moved with the breeze, framing a face so serene it made my chest tighten. The faint flush on her cheeks, the way her lips curved ever so slightly—it was as if she belonged to this moment, this city, more than anyone I’d ever seen.

For a moment, I couldn’t move. I drank in the sight of her, committing every detail to memory. Then, a voice called her from inside, and she turned, disappearing into the room. I waited, hoping she’d reappear, but the balcony remained empty.

Shaking my head, I forced myself to move. My steps were quicker now, my heart racing with something I couldn’t quite name. By the time I returned to my hotel room, the feeling had morphed into a need—a compulsion. I rolled up my sleeves, grabbed my charcoal pencils, and approached the easel I’d set up by the window.

The lines came easily, flowing from memory to paper as if she’d etched herself into my mind. The curve of her lips, the slope of her nose, the way her hair had danced in the wind—each stroke brought her closer to life. I worked with a fervor I hadn’t felt in weeks, adjusting, refining, until the figure on the page mirrored the one that had captivated me.

I stepped back, studying the sketch. She was there, but she wasn’t. I’d captured her image but not the essence—not yet. The fire was back, though, and I felt alive for the first time. In what felt like forever.

rubbed my eye, trying to wipe away the sleepiness threatening to overtake me as I leaned against the wall with the rest of my group. Whoever decided that 4 a.m. was a good time to fly needs to be smacked. Hard. My eyelids started to get heavy, and I began to nod off when a voice called over the loudspeaker announcing that first class for our flight could begin boarding.

I groaned and smacked my head against the wall, wincing at the dull throb that now pulsed in my temple.

"What's wrong with you?" my roommate asked, poking my side from the chair she sat in.

"I didn’t get much sleep last night," I grumbled, swatting her hand away and rubbing the sore spot on my forehead. In reality, I didn’t sleep at all. My mind kept thinking about the beautiful stranger in that photo. Was he a tourist? What was his name? Was he that beautiful up close? These were the questions that kept me awake until we had to head to the airport, where I now stood miserably as the wealthy boarded the plane first.

"Screw the rich..." I muttered under my breath, running both hands down my face. Once it was our turn to board, my group got in line. I handed the stewardess my boarding pass. She scanned it, frowned slightly, then scanned it again, her brows furrowing. A pit grew in my stomach.

"Oh no," I thought as her eyes darted over the words on her screen. She turned her gaze to me and spoke. "I’m sorry, miss, but it seems we double-booked your seat..."

I felt like I was going to throw up. My face must have turned even paler as she continued typing on her keyboard.

"But it looks like I can bump you up to business class, if that’s alright with you," she said.

My sleep-deprived brain barely processed her words. I just stared at her for a moment as she added, "Free of charge, of course, for the trouble."

I swallowed the lump in my throat and nodded, unable to form words. She printed a new boarding pass, handed it to me, and waved me through. I found our professor and gave him the update. He gave me a thumbs-up before pulling on his headphones and lowering his eye mask.

I made my way to the business class section, searching for my seat. It was in the middle row, meaning I had another seat next to mine. But when I saw the amount of space I had, I didn’t care—as long as my neighbor didn’t mind if I snored a little. I plopped down into my seat after putting away my carry-on. Letting out a sigh, I pulled the provided blanket over myself, turned to one side, and drifted off to sleep before we even took off.

I woke up to the hostess asking if I needed anything to eat or drink. Blinking my eyes, I regained consciousness and asked for water and a snack. Once I had both, I sat up a little and took in the seat I was in. It was almost like a mini cubicle with a small folding table and a TV screen. To my left was the divider between me and my neighbor, who was busy drawing something on a sketchpad. I couldn’t see his face clearly because of the mask, hat, and headphones he wore.

My attention shifted back to my snacks as I pulled out my laptop to get started on more editing. At some point, I fell asleep again but woke up before they came around to deliver more food.

"Thank you," I whispered to the steward, who smiled and nodded before moving on.

"That’s a beautiful picture," an accented voice said behind me. I whipped around to see my neighbor looking at my laptop.

"Oh, thank you," I said shyly, setting my food down. I tried to ignore the flutter in my chest from his compliment as I continued to work. Out of the corner of my eye, I noticed him shuffle and take off his headphones and hat, but I averted my eyes before he caught me peeking. To distract myself, I clicked on the picture of the stranger and zoomed in slightly.

My neighbor made a choked sound. When I looked back at him, his wide eyes were glued to my screen.

"Are you okay?" I asked. He looked at me, then back at the screen.

"That’s me," he said, pointing to the photo.

My gaze darted to the picture and then back to him, my eyes narrowing. "What are you—" But before I finished, he took off his mask, and the words died in my throat.

Here, next to me on this very plane, was the reason I couldn’t sleep last night. I sat there, mouth open in shock, as he looked at me with an odd expression.

"Are you serious?" I asked.

He nodded. "Why do you have my picture?"

"I wasn’t taking YOUR picture." I clicked to expand the photo. "I was taking a picture of the Eiffel Tower."

His eyes widened, and his cheeks flushed red. "Oh..." His voice was soft as he apologized.

I nodded, and we sat in awkward silence for a moment.

"What are you drawing?" I asked, trying to ease the tension, pointing at his sketchpad.

He blushed more, tilting the pad toward me. "It’s just a sketch of a woman I saw last night," he muttered, avoiding my gaze. The whole time we’d talked, he hadn’t fully looked at me. I wondered if there was something wrong with my face. But when my eyes roamed over the drawing, I tilted my head in confusion.

"She looks an awful lot like me," I chuckled, knowing that wasn’t possible. But his eyes snapped to me, then back to his drawing. His mouth opened in wonder.

"Were you staying at Hotel de la Tour Eiffel?" he asked.

My laughter died. "Yeah... How did you—" My eyes widened as we both looked at his drawing. It was me.

"Holy crap," I muttered as he shook his head in disbelief. We looked at each other again.

"I can’t believe I found you," we said in unison.

We spent the rest of the flight chatting quietly about everything and anything. Soon, we were landing, and a wave of sadness hit me. I’d never see him again. Once it was our row’s turn to exit, I grabbed my things and turned to him.

"It was nice to meet you, Hyunjin," I said with a soft smile before slowly making my way down the ramp and into the terminal. As I stood off to the side waiting for my group, I heard my name being called. I looked around to see Hyunjin waving at me.

When he reached me, he blushed, clearly embarrassed. "Sorry about that. I just..." He took a deep breath. "Can I have your number?"

I gave him a shy smile and nodded. The smile he gave me made butterflies flutter in my stomach, and I thought to myself, I might have brought home something better than pictures.

[fchsadfa]

"To build their model, Niu borrowed mathematical techniques from an unexpected source: general relativity, the theory used to describe the warping of space and time. While relativity explains how gravity bends space-time, the researchers applied similar geometric principles to explain how the looping paths of yarn create curvature in knitted fabrics."

"Where knitting meets origami

Niu explains that the work fits into the larger focus of Kamien's group, particularly its research on kirigami, the art of cutting paper to create complex, foldable structures.

"Kirigami, much like knitting, is an example of how geometry can be used to encode mechanical properties into a material," she says.

The team's previous work has explored how strategically placed cuts in a sheet can cause it to morph into specific three-dimensional shapes when stretched. The insights from knitting take this idea further, showing that a material's internal structure—not just its cuts—can dictate how it folds and unfolds.

"The parallels between knitting and kirigami are striking," Kamien says. "In kirigami, you add cuts; in knitting, you add loops. But in both cases, you're programming geometry directly into the material so that it shapes itself without requiring extra inputs like heat, hinges, or reinforcements."

Dion coined a new term for this approach: knitogami—a fusion of knitting and origami that captures the idea of self-folding textiles. "We call it knitogami because it extends the principles of origami into a soft, fabric-based medium," she explains.

"Instead of relying on folds and creases in paper, we're using the inherent elasticity and structure of knitted loops to create dynamic, shape-shifting materials." "

First of all, stitch patterns a and c look like smocking patterns (Canadian smocking, wave pattern and lattice pattern specifically)

Secondly, I need to mess around with stitch pattern E. That's wild

[whats-in-a-sentence]

In a simple cubic crystal, layers of atoms stack one directly above another, so that all atoms lie along straight lines at right angles, as figure 7.34 shows. (...) The unit cell of the primitive cubic structure is shown in the cutaway portion of figure 7.34.

"Chemistry" 2e - Blackman, A., Bottle, S., Schmid, S., Mocerino, M., Wille, U.

[cog5]

Brute Force Adventure Writing?

I’ve found success lately by writing my 1D6 tables as 1D8 tables, then cutting the two least interesting options. Even if I write just one extra option and make a cut, it tends to make a stronger encounter table. A 1D4 table? Write five options, review and cut the most boring or repetitive one. Write lots, keep what’s best. Is this brute force adventure writing? Or like just, normal editing? I don’t know. I’m not a professional. But it works.

In one of my latest dungeon rooms, players can encounter a giant urn full of nanomachines. The Nanites can combine into several forms. I went for a 1D4 table in this case, wrote five options, then cut one.

Nanomachine Gestalts

1. Statuesque Humanoid: From a distance, the swarm appears human. Tall and graceful, gleaming with reflected light. Up close, the figure takes on an alien appearance, its proportions exaggerated, its face grotesque.

2. Kaleidoscopic Lattice: Arrays of Nanomachines chain together, rapidly filling the room with a complex matrix of crushing fractals. Passage through the area becomes increasingly dangerous.

3. Electric Amoeba: An amorphous shape, its iridescent volume warps and quivers in all directions. Appendages appear and disappear as it blindly probes the room. 

4. Monolithic Geometry: Cubes and other angular shapes form a mass of intersecting surfaces, reflecting the world around it. New structures erupt like animated pyrite, extending to unexplored passageways, while trailing offshoots reconstitute into the primary locus.

5. Simulacrum: The Nanomachines appear as a cloud of iridescent dust. Several are inhaled by a member of the adventure party, entering their bloodstream and brain. The remaining Nanites will fill the room, creating a space based on the adventurer’s memory, for all to see.

I ended up cutting #3 from the available options. It was too similar to option #4 in form and function. Plus I wanted to push the “machine” aspect in nanomachine. Really, #3 inspired me to write #4 – a classic blob monster, but more mechanical? Of course. Giving the encounter a more organic bent could have been interesting? But I thought the humanoid form in option #1 filled that niche well enough. Given every other option, #3 didn’t have a lot going for it. Overall, I think the table is stronger without it.

And I just want to say…

Why’s it gotta be so hot? I’m so unproductive in the summer. Any amount of heat seems to sap my motivation. So, a perfect time of year to wrap up a mega-dungeon. And yet, progress has been made. I just cracked 200 pages on my latest project, The Electric Triptych of the Tetric Necromancer. By the end of next month I should have all 12 (out of 12) areas laid out and edited. After that, I’ll want to draft an introduction, a timeline of events, a few adventure hooks and such, but the entire dungeon will be playable from start to finish. That’s something.

Stay cool, people.