The Geometry of Magic
To accommodate the large crowd for explaining these concepts, using a blackboard wasn't feasible, so I set the coffin containing Isabel's corpse on the floor and raised her as a servant. Then, converting mana into water and that water into ice... I sequentially created points, lines, planes, and solids in the air, interfering with the atmosphere to keep them suspended without falling.
"When they realized that any concept-less point multiplies into a line with length, those lines layer into a plane with area, and that plane becomes a solid... mages vaguely understood. Shapes with an unknown n-dimensional concept must follow a similar process."
"…"
"But this alone doesn't establish hierarchy, so until now, we've recognized magic formulas through linearized equations."
Length, area, volume. These were mere metrics analogous to mana volume. To establish hierarchical order, we needed the cause behind these results—the proper causal relationship.
"Then... let's briefly return to the origin."
I cleared the line, plane, and solid, leaving only the point.
"A point with no concepts, a state without notions for understanding formulas. But like pointillism, multiplying these points creates a line with length, at which point we can recognize Keter-rank magic formulas..."
I enlarged the ice line and fixed it in the air.
"Why are this point and line separated into lower and higher concepts? Simply because length emerges? No. But not wrong either. It's merely a byproduct of the result."
"…"
"Precisely, a line with length is a state that completely transcends the point's time axis, becoming an omniscient existence observing all of the point at once... That's the correct view."
"Isn't the explanation too arcane and jumbled?"
I smiled wryly and nodded at the remark from Teor el Binheim Roswell, the Roswell elder seated at the back.
"I'll explain more simply from now on."
I colored sections of the suspended ice line.
"In the line's world, only length exists, so it can only be a red line or blue line like this. Movement is only left-right, so when lines with colors see someone..."
I rotated the ice line forward.
"They appear as points."
The wise ones probably caught on here. Lines cannot perceive themselves as lines.
"Adding the y-axis of time, the red and blue lines living on the x-axis will live and die by their will..."
Trailing off, I greatly magnified the section recording the red and blue lines' lifespans along the y-axis of time.
"This life becomes a zigzag of paint on canvas. Yes... in a word, a plane. On this plane, all starts and ends of lines are recorded at once."
Similarly, a plane with area and length flowing along the z-time axis becomes a volume solid.
In other words...
"The time of lower concepts is already begun and ended from the higher concept's view, observable at once. Meanwhile, the higher concept completely transcends the lower's time axis... This is the definition of hierarchy that lets us recognize magic formulas as points-lines-planes-solids."
Sipping the black tea Alphonse had prepared, I decomposed and erased all the ice illustrations.
"Questions? Raise your hands."
"Hmm, transcending the time axis is hard to grasp..."
Looking at Teor el Binheim Roswell, who raised his hand and questioned immediately, I created a large ice rectangle in the air.
"Assume that's canvas."
"Yes."
"And..."
In the center of the large ice rectangle, I delicately carved a small one.
"If that small rectangle becomes a volume-possessing cube, it can't exist in the canvas world. The moment it gains volume, it's ejected. That's transcending the time axis."
"So theoretically, we solids would be ejected if ascending to a higher concept?"
"Yes."
"Hmm..."
"From that higher existence's view, it could observe our births and deaths at once. Simply analogized as shapes:"
I drew a 4D tesseract and octachoron in ice, rotating them so everyone in the hall could see.
"Higher concepts' shapes can be expressed as cross-sections of current ones, but that loses intuitive advantage. For Terminus and beyond, recognizing formulas via equations as always will help."
A brief silence fell.
Terminus and beyond didn't exist. History's peak mage attainment stopped there.
"More questions? None?"
Scanning the room, Uriel raised her hand.
"...Speak, Professor Uriel."
"You said higher concepts observe lower ones at once? But we solids can observe all faces of solids."
"Precisely, only along time's flow, so incorrect. For a large die, observing all faces requires moving to see hidden ones or applying force to turn it."
"Use mirrors."
"A mirror's reflection is strictly a plane. Arranging many requires eye movement to capture each, introducing time flow..."
"Ah... I get it. Understood."
Uriel cut me off before I finished.
What a rude woman.
"Tch—"
I deliberately clicked my tongue loudly.
Her action was objectively rude, drawing unwelcome glances from around.
"…"
Uriel flushed, hunching her shoulders.
That should suffice.
After a few more questions, I ended the lecture.
"I-it's... done...!"
As Ferzen left the hall post-lecture, surrounding mages deliberately cast spells, successfully recognizing their formulas as points-lines-planes-solids, and cheered excitedly.
More intuitive formula recognition made countering or altering them easier—who wouldn't rejoice?
'…'
Uriel too, though miffed, subtly recognized a nearby spell that way and clenched her fist tightly.
Hiding joy was her pride.
'More than that...'
She regretted not asking one more thing.
What could she do? She'd acted rudely in his prickly state—self-inflicted.
Clack-clack, heading from the hall to the main building, Uriel climbed the ramp to her office.
The guests were prominent, but her condition wasn't great for prolonged greetings.
'...In?'
Next door, Room 404. Ferzen von Schweig Brutein's office.
Shifting steps, Uriel gripped the knob, hesitated... but curiosity won.
Knock-knock.
Hearing his faint voice, she entered.
"Um..."
Thanks to her lecture rudeness, though invited, he didn't glance—cold as ice, making Uriel flinch.
"Sorry. My condition's worse than thought, so I was unintentionally rude in the hall."
"Using illness as an excuse is quite convenient thinking."
"Then you do the same..."
Pouting, Uriel pulled her brimmed hat low.
He'd crippled a 12-year-old girl without apology—hypocrite.
But she swallowed it.
"So why are you here."
"From what you said, transcending the time axis lets lower concepts ascend to higher."
"Correct."
"Then extreme acceleration would..."
Ferzen nodded, grasping her point.
This world's people dimly knew speed-time correlations.
"Yes, at overwhelming acceleration nearing—reaching—light speed... we'd be ejected from this world."
"Ah..."
"But massed objects..."
"Can't exceed light speed, so impossible now."
"You've developed a habit of interrupting. Uriel."
"Don't call me so familiarly."
Uriel scowled, glaring.
But Ferzen ignored it blankly, propping his chin mockingly.
"Curious about this when your body couldn't withstand such acceleration without dying—surprising."
Glancing briefly at her vulgarly prominent chest today, Ferzen smirked, rising.
He blurred focus viewing others intentionally, but that strained his eyes... jiggling large breasts urging refocus? Torture.
Unaware, Uriel flushed at approaching Ferzen, pointing accusingly.
"Y-you really! Every time you look, such sexual harassment! Don't you have designs on me?"
Like boys bullying crushes, his childishness irked her, she added.
"If I did, I wouldn't have wrecked the engagement."
A mere step away, Ferzen pushed her brimmed hat down.
"Overinflated ego, Uriel."
"D-don't touch...!"
"Who blocks the door saying that."
"…"
Fixing her hat, Uriel stepped aside.
Ferzen exited, eyeing her still inside.
"Not coming? What'll you do in an empty office."
Flinch!
At his word, realizing her oddity, Uriel dashed out, closing the door—but pain from jiggling breasts dropped her groaning.
"Ow... sorry, close the door for me..."
"So demanding."
Ferzen shut it amusedly, activated quantification, leveled the knob horizontally.
Descending to the first floor, he checked the department office for corpse delivery schedules for students, then headed to the parking lot.
'Time...'
Nearing 10:30; arrival around 10:50.
Lunch, bed with fertile Euphemia, embrace her under warm sun-blanket...
Perfect cozy day.
"A-all prepared."
Sleeves and skirt frills cutely adorned, floral embroidery on pale pink lolita dress. White accents harmonized with her fair skin, evoking fresh serenity.
Sun strong today, so hatted and parasolled Euphemia shyly held a sandwich basket in her free hand, calling maids.
After touring capital with Ferzen...
Intrigued, she'd heard Tisbe River ferry rides were a capital highlight—today's plan.
Sandwiches for simple lunch admiring scenery aboard.
Maids approached; Euphemia exited, descended stairs to sunny garden.
"Ah..."
Strong fresh breeze from behind lofted her hat, dropping afar.
"…"
But retrieving it wasn't maids—it was Ferzen, home unusually early.
"Going out?"
"Y-yes..."
Fatigue-weary face. Faint scowl screamed stay put, no nonsense.
But she'd prepared sandwiches diligently...
Grip.
Euphemia mustered courage, clutching his hem.
"P-prepared... d-diligently..."
"…"
"J-just a bit out..."
"…"
Lifting basket desperately, but his silence signaled refusal usually...
Unable to continue, she resignedly dropped hands.
"Go and return."
Yet he gently hatted her, permitting; Euphemia widened eyes, bewildered.
"What? Expected scolding to stay confined?"
"…"
"I know what I raise isn't a pet dog but a person—no intent to stifle."
"Th-thank you..."
Awkward honorific.
"But dogs wag tails greeting owners joyfully..."
Hinting, trailing off.
Euphemia fidgeted long...
"W-welcome... back..."
Soft murmur.
"Yes. Return before sunset."
Her tension unfounded, Ferzen passed into mansion.
She watched his fading silhouette beyond closing door...
Standing long.