circle graphs

Circle packing theorem

Recently I heard about this surprising theorem. I already knew about Fáry’s theorem: if a simple graph can be drawn on the plane without crossing edges (i.e. if the graph is planar), then it can always be drawn in such a way that all edges are non-crossing straight lines. Hence, restricting to straight-line edges doesn’t give a strictly smaller class of planar graphs. I found this to be quite surprising already.

However, much more is true: you can always draw a simple planar graph in the plane in such a way that you can place non-overlapping circles on the vertices, such that two circles touch if and only if the corresponding vertices are joined by an edge. In particular, the drawing clearly satisfies the result of Fáry’s theorem. This result is called the circle packing theorem.

Epitome of Eighteen Histories: Kururi Orihara & Mairu Orihara Episode

This is a story from the past

A twisted story from the past.

A Few Years Ago   Raira Junior High

“Liike I saiid~, butler clothes fit Yuuhei Hanejima-san the best!”

“……Deny (No) …… Alcohol (bartender clothes) …… Fashionable (looks the best)…….”

“Noo, stupid, stupid Kuru-nee! You can drown in a cocktail shaker for all I care!”

During break the first year sisters Kururi and Mairu were fighting over an extremely meaningless topic.

“Both of them are similar though, right?”

The boy who stated that not reading the mood was glared at by the two of them with a look that said “this guy doesn’t even know anything…..”, and although receiving a shock he has doubt over his own disposition with his heart throbbing at the strangeness of those cold gazes, but that is an entirely another matter.

“But Yuuhei Hanejima would like both of them, right?”

When a girl nearby said that, the two of them nodded at the exact same timing.

“Of course!”

“…..Of course….”

Hearing that, their classmates laugh in amazement.

“Really, you two get along well, don’t you.”

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More Crocodile wallpapers! (1440x2560) Also a little something that had me laughing for quite a while! Enjoy!

anonymous asked:

Yo, short ass sentence prompt. High school au cat the cheerleader and Kara the jock. Go! NC x

“Why do you pretend to be so mean to everyone?” Kara asked without looking up from her advanced mathematics book. Cat narrowed her eyes at the words, scratched out a circle in her cheerleading graph before adding something to it and the tip of her bright red pen almost ripping the paper. “Because, I’ve noticed being a rich blond girl in highschool isn’t exactly a sinecure and I am determined to thrive and to be remembered for it. This is why I chose to appear strong and unmovable, bitchy even. I don’t want people to see me as soft and caring.” This time, Kara looked up from her textbook and Cat felt her heart race into her chest at the sight of the bright, bright smile etched on the other girl’s lips. “I see you as soft and caring and I respect you, maybe more than all the jerks you’re hanging around with.” The kindness in Kara’s voice melted Cat’s reason and she finally smiled too. “I know. I still don’t understand how exactly you managed to get through my walls but … I’m glad you did.” She admited, her voice so low and small she doubted Kara heard her. 

The smile on Kara’s face, bright as a thousand suns and the love, pure as crystal water, swimming in the ocean blue eyes let her know she’s been heard, loud and clear. 

Search (Jongdae x Reader), pt. 1

Kim Jongdae: best friend and little shit.

Part 2 ○ Epilogue

Request: Would you be willing to write a soulmate AU with a member (you can pick!) of EXO’s beagle line? Thank you so much ❤️

fluff, 2.3 k words, jongdae/reader, soulmate au

// AGE 6

“________, this is Jongdae.” Your mother lets go of your hand and gestures at the boy in front of you, his hair tousled and his lips curved in a small smile.

You offer him a shy wave, which he returns.

“Aww,” Jongdae’s mother coos. “You two are so cute. Play nice, okay?” With that, the two mothers leave you two alone. You watch them close the door to the playroom, which is lined with toys and games of all shapes and sizes, before you turn back to face Jongdae.

“Hi,” he says, sweet and playful.

“Hi, Jongdae.”

He gives you a toothy grin and extends a pudgy hand out toward you. “From now on, let’s be friends,” he says, decisively.

You grasp his hand, the kind of bright, innocent smile on your face that only six year olds can truly muster. “Sure.”

// AGE 14

“Hey, Jongdae, are you gon—“ You stop talking, dropping your bag on the ground in shock, when you see Jongdae sitting on the floor eating a bag of potato chips. Your bag of potato chips.

He quickly turns around, eyes widening upon seeing you, and then he’s gone. The little shit scrambles up and away, socks sliding hazardously against the hardwood floor.

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