Rithmatics: Part 4, Obtuse Triangles

Rithmatics: Part 4, obtuse triangles

In the previous posts we have addressed all acute and right triangles.  In this post, we look at what happens if the triangle is obtuse.

Obtuse triangles and the 9-Point Circle Construction

In an obtuse triangle, two of the altitudes fall outside of the triangle. This appears to be a problem, but we can work around it.  The 9 point circle construction we have been using so far is the special case of a more general 9 point conic construction that starts with 4 points.  This more general construction produces a circle whenever the 4 points are the three vertices of a triangle and its orthocenter (the point where the three altitudes intersect).  To find the orthocenter of an obtuse triangle we have to extend the altitudes to find where they intersect outside of the triangle.  We then use the three midpoints of the sides of the triangle, the three points where the altitudes intersect the opposite side (or side extension) and the midpoints of the segments connecting the orthocenter to the three vertices of the triangle.  As you can see in the diagram below, this ends up being the same triangle you would get from considering the acute triangle formed by the orthocenter and acute vertices of the original triangle.  This means that obtuse triangles can give us a different perspective on our circles, but will not produce any new patterns we couldn't get using acute triangles.  The advanced rithmatic theorist should be aware of this but for basic rithmatics it is fine to ignore obtuse triangles.

Rithmatics: Part 4, Obtuse Triangles

More Posts from Kalynaanne and Others

9 years ago

We know kandra can imitate animals - TenSoon is particularly great at using his dog bones to his advantage. But you know what I want? Kandra birds and in particular Kandra vultures 

(BoM spoilers under the cut)

In Bands we find that we are entering the age of flight. This clearly means that we need a kandra who realizes they should learn to imitate a bird so they can keep up. VenDell tells us that small animals (like bunnies - thanks Wayne - also I really want to know more about “that book”) are extremely difficult because they “need a certain mass to hold [their] cognitive functions.” This probably rules out small song birds, but there are large birds of prey that might be big enough. One of the largest birds (discounting ostriches and emu that can’t fly) is the Eurasian black vulture that can get up to 30 lbs. I really like the idea of a kandra vulture. Can this please happen?

Then we can get the kandra vulture who has an intimate understanding of flight because they have eaten birds and absorbed their anatomy and learned to fly to hang out with Ranette and Sophi Tarcsel and create all kinds of wonderful things?


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10 years ago

In the chasms in Way of Kings, Dunny creates an insult poem by making the rhythm fit the song Mari’s Two Lovers. Rock then goads Dunny into singing it before joining in.  Ashiok and I have been working on lyrics for the song.  This is what we have so far, but this is a Rosharan drinking song and we are pretty sure it has many, many more verses, so we are releasing it into the wilds of the cosmere fandom to see if anyone else happens to have any verses to add. Current lyrics under the cut.

Dunny’s poem:

Hey you big buffoon, 

you smell like a wet hog,

so go out by the moon, 

And jump yourself in the bog.

Mari’s Two Lover’s (lyrics by Kalyna and Ashiok)

Gather round and hear

this tale I have to tell

of our young Mari dear

and the men she loved so well

Mari was a lass

Though Not the brightest sphere

She often cut her class

to a wander far and near

There she met a lad

who took her out for wine

But she arrived with Khad

and left with his brother Stine

Khad and Stein were twins

who looked  so much the same

it was beyond her ken

that the two had different names

The twins had a tell:

two eyes of different hues

they were distinguished well

cause their eyes were switched, it’s true

Their town had a dance

And Mari begged to go

While there she whirled and pranced

Always watching for her beau

As she spun around

She fell/came into Khad’s arms

She was extremely proud

Khad had fallen for her charms

Then she came to Stine

And danced with him a turn

She stared into his eyes

Oh but still she didn’t learn

Khad had brown and blue

While Stine had blue and brown

But our dear Mari knew

Her beau was the best in town

Over in a lait

An axehound pup bit Stein

To keep up the charade

Stein cut khad with some fork tines

Now this song must end

I sadly have to go

But Mari and her men

Oh their tale goes on you know


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10 years ago

Hey y’all, it’s Kaly reads a poem time again.  This one is called Entropy’s Embrace.  It isn’t quite a cosmere specific poem, but it is a great poem, was written by our friend worldsingers and it does have some lovely cosmere evoking imagery.  You can find the text and a review of the poem here.


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10 years ago

brightlord-kholin replied to your post: Hello and welcome to Cooking with Kaly...

I’ve gotta find a way to substitute the meat with a vegetarian fake meat thing because this looks delicious and I need to try it.

Hmmm... I would probably start by playing with a falafel recipe, but  I could also see tempeh working well. Fake meat is also a decent interpretation of “soulcast meat,” so you can even claim you are sticking to canon :D.  If you decide to try it, I would love to hear how it goes!


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10 years ago

Shallan's Journey: Chapter 4, In Which Pattern Gets to Play

This morning started as the last few had, with Shallan meeting the others for breakfast and then returning to the library to study English with the girls while Molly tried to wrangle the boys into helping her clean the house. Today, however, there was something going on that called Molly out of the house and left the kids to their own devices. Sirius was still home, of course, but he spent most of his time upstairs moping with Buckbeak and when he did come out was rather more likely to cause mischief than to be a responsible adult. As the oldest Weasleys left in the house, Fred and George had decided this meant they had free rein to do as their hearts desired. Naturally, this meant that Ginny had bright green hair and Hermione's pens were dancing away from her every time she reached for one. A few minutes ago, Hermione had finally snapped and begun laying into the twins about how irresponsible they were being and trying to insist that they set everything to rights and then leave them alone. Ginny had joined in and now there was a full scale argument making it difficult for Shallan to think. She narrowed her eyes as she contemplated the best way to handle this.

Shallan had been reluctant to use any of the moderate supply of stormlight she had brought with her in case she ended up really needing it later, but she decided that it probably wouldn't hurt anything to use a small bit of it. She flipped to a clean page of her notebook and drew out a fierce image of Molly shaking her finger along with a few other quick sketches that would allow the image to move. “Pattern,” she whispered, “would you like to startle them?” Pattern buzzed excitedly in response, but Shallan quickly shushed him. “We are going to sneak out into the hall where I'm going to attach an image of their mother to you and then...” she continued to whisper instructions as they sneaked around the argument and out into the hall. She infused Pattern with stormlight and triggered the illusion before quickly slipping back into the room to watch the results.

She had just gotten back to the table when Molly's voice boomed from the door, “Fred! George! What is the meaning of this?! I told you to leave the girls alone!”

Everyone immediately froze and looked toward the door while Fred and George jumped away from the others, “Mum!” “We didn't know” “you were” “already back!” “Of course” “we were” “being nice” “to the girls” “anyway, but...” they trailed off in confusion as the image of their mother flickered out and died and they noticed Shallan trying and failing to hide her snickering.

Ginny was the first to figure out what must have happened, “Whoa! You can do wandless magic? That was a really impressive illusion! How did you do it? What else can you do?!”

Shallan shrugged. She was getting much better at understanding English, but was still shaky when it came that fast and still not particularly confident speaking it herself. “It is... not your magic... Ummm...” she pulled out the sphere she had already pulled some of the stormlight from. “I am a...” she searched briefly for an appropriate English word and gave up “lightweaver, so I use these,” she held up the sphere, “to... to make things.”

Hermione began bouncing on her toes, “Oh! But that's fascinating! May I see it?” She held out her hand in question. Shallan handed the sphere to her and Hermione examined it carefully. “Oh wow. That looks like an emerald! And it glows... Can you show us how you do it?”

Shallan nodded, “Sure,” as she took the sphere back, “um, what should I make?”

Ginny grinned at her, “Can you show us your boyfriend?”

Shallan gave her a slightly sad smile and turned to her notebook to draw out the sketch of Adolin that she would need for the illusion. She breathed more of the stormlight in from the sphere so that she was just slightly glowing and breathed it out as she created an illusion of Adolin standing there grinning at them in his impeccable Kholin blue uniform. Storms. She had been so caught up in dealing with everything that, even when telling the girls about people from Roshar, she hadn't realized how much she missed him. Seeing her illusion standing there now though... It looked just like him, of course, except that the storming man never stood that still...

She sighed and buried her head in her arms on the table as she wondered what Adolin and Jasnah and the all of the others were doing. She knew Jasnah wouldn't give up on her, but, storms, she didn't know where she was, how did Jasnah have any chance? A gentle touch on her shoulder pulled her back and she looked up to see that Ginny had slid into the seat next to her and was now giving her a concerned look. Shallan didn't have the English to fully explain, so she just shrugged as she whispered, “I miss them.” She let Ginny pull her into a hug as she put herself back together. It wasn't easy, but then nothing in her life ever seemed to be.

The others looked on somewhat helplessly as Ginny comforted Shallan. Hermione desperately cast about for a distraction. She wasn't good at these things like Ginny was, but she did know that her own usual response to being upset was to throw herself into a project. From her interactions with Shallan over the last few days, she figured there was a good chance that Shallan was the same way. Her eyes flew around the room for inspiration and finally lit on the wand that Fred was holding loosely at his side. “Oh! Of course!” She blushed as everyone immediately turned to look at her, “Well, it just occurred to me that we know Shallan has something like magical abilities, but we haven't done anything to test whether she can use our magic.” She looked beseechingly up at Fred, “Could Shallan borrow your wand for a minute, just so we can see if she can use it?”

Ginny rolled her eyes at Hermione,“Really? You think this is what we should be doing right now?” In the mean time, Pattern had buzzed over to Shallan and explained what Hermione was suggesting. Shallan gave Ginny a small smile, “It's ok, I want to know too,” and then looked eagerly over to Hermione. This was good. As long as she didn't think about home she would be fine. Pattern would probably fuss at her later for having too many lies, but it wasn't like there was anything productive she could do.

Fred shrugged curiously and handed Shallan his wand, “Just be careful with with.” Shallan nodded seriously as she reached to accept it, “I will.” When her hand closed around the handle, her eyes widened and she stared down at the wand. She swore she could almost feel it vibrating. “Pattern, what is this thing? It looks like a fancy stick, but...”

“Mmmm, it says it is a wand.”

“Well, yes, that is what they told us, but what is it?”

Pattern seemed to shrug. Shallan could never quite figure out how his twisting mass of lines mimicked human gestures, but he managed it some how. “It says it is a wand.”

Shallan rolled her eyes. Sticks, even fancy, magical sticks, were hopeless. She looked back up at her new friends. “Sorry. I was...distracted.”

Hermione smiled, “That's ok. Do you feel anything from the wand?” Shallan pursed her lips, “Yes, but I don't have the words...”

“Ok, well, let's try something. Your magic system seems to involve light, so let's start there.” Hermione grabbed a pencil, “You move the wand like this and say 'lumos' and the tip of the wand should light up.”

Shallan tried unsuccessfully. She gave a slight sigh. There was no promise this was going to work, and even if it did, she shouldn't have been expecting it to work on the first try. George pulled out his own wand and stepped over to her. “First off you should be holding the wand like this,” he held his wand out in front of him so that she could see and imitate. “Then you move it like this.” He demonstrated the motion and the encouraged her to do it with him, “You don't want to whip the wand around too fast, but you do want a nice smooth motion.” They practiced the motion a couple of times before George was satisfied. “Now, when you get to this point in the motion, you say 'lumos'”. George demonstrated and the tip of his wand lit up. Shallan concentrated as she tried again and, to her delight, the wand tip actually lit this time! Interestingly, casting the spell almost felt like pushing stormlight into something...

She grinned before narrowing her eyes briefly at the wand in contemplation and then laying it carefully on the table. She grabbed her sketchbook and drew out a quick sketch of an axehound. She picked the wand, still lit, back up and, imagining that the light on the end of the wand was stormlight, pushed it out into an illusion of an axehound.

Shallan's whoop of victory was drowned out by Ginny's shriek. Even though she knew on some level that it was just an illusion, she leapt backwards and her voice trembled slightly as she squeeked out, “What is that thing?! It looks like something Hagrid would love...” The twins examined the new image excitedly and Hermione looked it over cautiously. Shallan grinned at them, “It's just an axehound. My brother,” her voice caught slightly, but she pushed it away and kept going, “my brother used to raise them. We keep them as pets back home.”

Having recovered from sudden appearance of the strange creature, Hermione looked back up at Shallan with eyes shining in excitement, “Regardless of what it is, you made it with Fred's wand instead of one of your glowing marbles! We have to get you a wand!”

At this point Ron wandered into the room, “What is all the screaming about?” Ginny gave her brother an unreadable look and pointed at the axehound in the corner. “Oh bloody hell!” He scrambled back towards the door. “That thing has way too many legs! Did Hagrid stop by?”

Ginny smirked at him. Her brother's reaction helped her get past her own, “Nope. Shallan used Fred's wand to create it. It is apparently like a dog where she comes from.”

Ron shook his head. “No way anyone would keep something like that as a pet.”

Ginny rolled her eyes, “I know we wouldn't, but if you grew up with them they would seem normal.”

“If that thing is normal, her world must be completely creepy.”

“Ron! Shallan is right here! If you can't be nice you can just bugger off again.”

Ron threw his hands up in the air and grumbled as he stormed off, “The crazy alien girl shows up with creepy creatures and suddenly she is all anyone cares about....”

Ginny shook her head as she turned back to Shallan. “I'm sorry about my brother. He isn't good at accepting new things.” Shallan nodded with a sigh. “I understand. I also have several brothers and, umm.. they are not always exactly...” she trailed off. She did not want to try to explain her family.

Hermione cut in, “Let's go find Sirius. Maybe he will know how we can get Shallan her own wand.” She led them farther up the house than Shallan had yet been. “You've met Sirius at dinner a few times, but he tends to keep to himself. I don't think he's very happy here.”

As they entered the room, Shallan's eyes were immediately drawn to the large creature in the room. Hermione had immediately turned to talk to Sirius, but Ginny caught Shallan's arm before she could rush over to it. “Whoa, Shallan. Buckbeak is friendly, but you have to meet him properly. Like this.”

She walked slowly until she was in Buckbeak's direct line of sight and then bowed. Buckbeak blinked at her and returned the bow, so she stepped in to stroke his neck.

While Shallan was watching this, Pattern buzzed up beside her “Is that a horse chicken?”

“I don't know anymore than you do, but it certainly looks like it. Storms. And they thought my axehound was strange...” Ginny gestured for Shallan to come closer and bow. It took a little longer this time for Buckbeak to respond, but he eventually did and Shallan stepped closer to join Ginny in stroking his neck. “Ginny, what kind of creature is Buckbeak?” Ginny looked at her slightly surprised, but then shook her head, “I guess you wouldn't know, would you... Buckbeak is a hippogriff.”

Shallan frowned as she examined him. “He can't be happy in this room. Does he even have space to stretch his wings?” Ginny sighed. “I think so? The reason he is here is complicated.” Shallan was distracted from her thoughts by the sound of her name.

“Hey, Shallan!” Hermione called. Shallan promised Buckbeak she would be back later, he was a fascinating creature, and walked over to Hermione and Sirius. “Can you show Sirius your axehound?”

She still had Fred's wand so she nodded as she carefully lit it with Lumos and then pushed the light from the wand into the image. Behind her she heard Buckbeak snort and Ginny calm him.

Sirius raised his eyebrows. “That's a mighty unusual way to use magic.”

Shallan shrugged. “It works.”

“Ah, that it does.”

Hermione cut in impatiently, “Do you see why we need to get her a wand? It's silly for her not to have one when she is capable of that!”

Shallan looked at them, “How would I get one?”

Sirius frowned at her, “That's the question, isn't it. I have more money than I know what to do with, so that's not a problem, but people usually get a wand by going to Ollivander's, and I'm not sure how we are going to get you to Diagon Alley. I'd take you, but Dumbledore won't let me leave the house.”

“Why not?”

“I was accused of a crime I didn't commit. You see...” he proceeded to explain the mess his life had been since the decision to make Peter the real secret keeper. At the end of his story he shook his head “I don't know why I'm telling you all of this. It can't be very interesting.”

Pattern buzzed appreciatively “Mmmm... so many layers of lies...I like this one...He would fit with your guards back on Roshar.” Shallan laughed helplessly at Pattern's comment. “He would, wouldn't he...Oh why do I keep finding myself in this kind of position...” She looked back over to Sirius. “I, ah, might be able to help with a disguise that would let you go out without being caught.”

Sirius's eyes lit up at the idea of being able to escape the house, but Hermione cautiously chimed in, “We'll have to make sure it holds long enough and isn't easily dispelled.”

Sirius wilted when Shallan nodded, but looked back up slightly hopeful when she added. “Yes, but we can test it. If we have enough light, I think it could work.”


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7 years ago

Your questions got me thinking and realizing that I have additional questions:

Szeth is likely going to summon his spren as a sword eventually. Is it going to make Nightblood jealous? Will they see it as a way to destroy twice as much evil? Szeth dual wielding blades? Or Nightblood getting frustrated because Szeth really needs a long spear for a sky battle right now and he is using the sprenspear instead of a sword?

Or, would Szeth summon his spren as a shield while using Nightblood as a sword?

What is the conversation going to be like when Nightblood and the spren talk?

Or, y’all, Szeth and Nightblood and the spren in a sort of threeway poly nahel bond. Maybe the bond could dampen the spiritual cracks in Nightblood so that they don’t automatically suck all of the investiture out of anyone holding them? 

I wonder what it’s like to be Szeth’s spren.  Like, the only thing they have to do is stick close enough that he can surgebind.  Nightblood takes care of everything else.  Did some Highspren just go: “This is gonna be the easiest assignment ever heck yeah” and then sleep through the rest of the book?  Are they annoyed at not having anything to do?  Were they tired of working with more conventional Skybreakers?  What do they think of Szeth siding with Dalinar over Nale?

I have questions guys


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10 years ago

Rithmatics digression

We've talked about 9-point circles.  Here is a further exploration of 9-point conics in case you are curious:

Rithmatics Digression
Rithmatics Digression

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10 years ago

Dear Cfsbf, 

It has come to my attention that the current logo used on the Coppermind (our wiki for all things Sanderson) was intended as a stopgap measure when it was added five years ago. We are planning to finally fix this and to do a redesign of the main page of the wiki. If you would be interested in helping/being part of this conversation (you don’t have to know about wikis), please reply to this post or shoot us an ask over at harmonys-coppermind :-) Any interest?


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10 years ago
So... I Was Experimenting With The Watercolor Brushes In FreshPaint And Playing Around With Ways To Get

So... I was experimenting with the watercolor brushes in FreshPaint and playing around with ways to get different effects for skies...and then this happened... so here, have a happy playful skyeel flying through the sunbeams with its little fishy spren friends.


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kalynaanne - Thoughts and Creations
Thoughts and Creations

So. I found my way to tumblr when I first discovered Brandon Sanderson's books. As a result, this, my main, was all Sanderson all the time. Tumblr won't let us change which blog is the main blog and my brain won't let me make this blog more general, so you'll find my general tumbling (currently including a great deal of Imperial Radch and Murderbot) on my "side blog" RithmatistKalyna.tumblr.com .

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