Making a sphere is something that comes up a lot in the realm of crocheting stuff. But how do you make a really good sphere?

This is something that I pondered one day. After a very brief and half-hearted go of it with google, I decided to answer the question myself… by throwing some math at it. How do you create a sphere when you are working in a row-by-row system (like crochet) where all you can alter is the number of stitches per row?

Well I assumed that each row is equal to a fixed polar angle , Δθ, within the sphere. At each θ, the circumference of the circle that is traced out within the sphere is proportional to sin(θ) (this was just a quickie derivation). Below, is a plot of what the circumference (y-axis) looks like for each θ (x-axis). So, to completely recreate this, I would want the number of stitches in each row to scale to fit this plot:

So far, so good. In the past, I have only ever used one method to make spheres. It’s fairly common because it’s really easy and, once you stuff it, it makes for a decent sphere. But… it’s not *ideal*. This method has you increase by the same number of stitches every row until you hit the middle bit, where you work rows even for a little while before you start decreasing by the same amount for each row.

Problem: this is what this method looks like when plotted against the ideal case (the blue dots represent the old method):

It’s just not the greatest. So what I did was to figure out what the ideal number of stitches per row would be, given the sin(theta) dependence. Result: I can make an ideal sphere for however number of rows I want. It’s not as clear as the tried and true method and it’s not intuitive, but it makes a nice looking sphere.

Here is a pdf file where I’ve written up the pattern for ideal spheres of different sizes. I haven’t tried all the different sizes out myself (since I would up to my eyeballs in spheres) so they *should* all work out nicely. Let me know if they don’t.

Enjoy!

p.s. Math rules!

**UPDATE**: Here is a link to a newer post where I include the pattern for some even bigger spheres! Embiggen!

***03.25.10 Update: An error has been spotted on the 10 Row Sphere, rows 5 and 7. (Oops – sorry all!) So this mistake has been fixed and the link has been updated.

***03.27.10 Update: Oops#2 located on 16 Row Sphere and is now fixified.

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Oh. My. I would love to scale this for knitting, too! — as the row gauge for a knit piece will not be the same as the row gauge for a crocheted piece of equivalent stitch gauge. This much I know. The plotting of “X” and “y” axises, however, are just hopelessly vague recollections from long-ago High School Calculus and so I despair of building on your genius…

(but I can’t wait to incorporate the crochet version into this year’s felted xmas decor!)

I agree, lisab! It would be great to figure out an ideal sphere in knitting too! Learning to knit is on my to-do list… so maybe someday I will try to figure that one out!

I just found you through Ravelry and was so impressed with your sphere! I can’t wait to try one out! Thank you for sharing!

This is wonderful! Here I was thinking that I was the only one who thought the other way of making spheres comes out skewed and not circular at all. I am thinking of making this the base for some spherical patterns. Thank you.

Thanks!! I’m glad you like the pattern and I hope it works out with your spherical patterns!

There are a couple of knitted perfect spheres. The best-looking one is by Brent Annable, at rav link

http://www.ravelry.com/patterns/library/stockinette-short-row-sphere .

hi! um.. i couldnt get the pdf pattern….can u plz post it on ur blog ??

oh, my! I totally agree… math rules!

I’ll try this pattern today for a baby present I’ll give this afternoon… hope to post something tomorrow…

😀

This PDF is amazing! I’m using them to make a molecular structure and they’re coming out beautifully.

Forgive me if this has already been asked/addressed but I didn’t see it brought up – do you have any sort of copyright or ownership restrictions with these patterns? As in, if I want to use some of these with a design, would I be allowed to sell that pattern or items made with the pattern?

Thank you thank you thank you for this!

I’ve been using the “tried and true” method, but have

alwayshad trouble getting it to turn out curved without much convincing (and tugging and stretching).The first time I tried the 20 row pattern, it came out perfectly curved- no finagling required!

[claps hands with glee] ^_^

Oh this so rocks! I love you for posting that pdf! I’m trying to make an oddish and the sphere pattern I’ve got for it just turns out nasty. Your spheres are beautiful. I’ve made a Bob-omb, a Voltorb and soon an oddish out of your spheres. When I take a swing at my Despicable Me minion, I might just take one of your spheres and make it a capsule.

With the Voltorb, since he’s two colors I cheated a bit. At the center round I finished off and did the second half using the increases again and simply whip stitched the top and bottom together. It came out lovely. 🙂

Wow…you are a genius! I’ve always wondered why my spheres came out looking so wonky! Thanks so much!!!

Any way to work out the math for a GIANT sphere? I’ve never seen someone crochet a huge one and I’m wondering how it would go.

I would also love to know how to make a gigantic spere! Maybe 10-15 inches diameter?? I’m making an Umbreon for a friend, and want to package it nicely within a big hollow pokeball… any suggestions?

Thank you for this pattern!

Thanks for the sphere! Math rocks and so do you! 😀

hey try saying it as an odf not a pdf because i couldn’t download it because of that.

i mean saving.

not saying

this is AMAZE-BALLS!! what a beautiful sphere! i do a lot of amigurumi and got tired of being able to see the increases and decreases! i can’t wait to try this pattern 😀 THANK YOU in advance!

This is so cool – the worlds of mathematics and crochet collide. I think this together with your crocheted holy hand grenade make you my new crocheting hero 🙂

You are my new hero 🙂

i’m wordless… awesome! Respect your dedication and patience 🙂

Thanks so much for the pattern! I made some Dragon Ball Z balls with it… http://queenoftheplushies.blogspot.co.nz/2012/05/dragon-balls.html

Love love LOVE your work 🙂

thanks !!!

Thank you for your pattern. New to this. What does Row 10) 4 scdec.mean?

Hi! 4 ScDec means Single-Crochet-Decrease the next two stitches together. Repeat this four times.

I feel terribly dense admitting this but I’m having the worst trouble trying to understand the pattern for the 12 row sphere. Once I hit row 3 I became befuddled. Could someone clarify what it means? Just the structure of the instructions. For instance, when it says “1, inc, 2, Inc, 1, inc, 2, inc, 1”, does that mean increase once, sc twice, inc? Sorry, I am quite new to crocheting.

Hi! Check out the top of the pattern for a break down of the lingo. “Inc” means “2 Sc in next stitch”. “1, inc, 2, inc, 1, inc, 2, inc, 1” means “Sc in next stitch, Inc, Sc in next 2 stitches, Inc, Sc in next stitch, Inc, Sc in next 2 stitches, Inc, Sc in next stitch”.

Hope this helps!

Oh thank you so much! 🙂 on with my pokéball 😀

Hi

I have a problem, I don’t understand if I have in row 9, 12 stitches and I do 4scdec as you explained, I don’t understand what to do with the final stitches , could you please explain what this: “Weave loose end through each stitch in the opening” means exactly. Thanks.

I think you’re looking at the final row of one of the spheres? When I say ‘4 ScDec’ it just means to ScDec 4 times. This will not use up all of the stitches from the previous row, but that’s ok. With the loose end of the yarn, weave it through each of the last 4 ScDec stitches and pull the yarn to close the opening.

Hope this helps!

Great, now I understand, Thanks a lot, I going to translate into Spanish if you aloow me to do it. maybe i is translated yet…I don´t Know and many people ask me. Fantastic work

I’m glad I read these comments! I’m currently on round 26 of the 26 row sphere, where it says 4 scdec around, leaving me with four unworked stitches. I was wondering what to do. Thanks!

Sorry but I’m still a bit confused. The row before the 4ScDec has 12 stitches. After the 4ScDec 8 of the 12 will become 4 stitches leaving 4 stitches. Do you mean weave these last 4 stitches together that are left? (You said last 4ScDec which is confusing as they haven’t been decreased.

Also, I’m wondering why you don’t ScDec all 12 to make 6 stitches and then weave these together so it mirrors the start.

Thank you. Awesome pattern 🙂

Today, I went to the beach with my kids. I found a sea shell and gave it to my 4 year old daughter and said “You can hear the ocean if you put this to your ear.” She put the shell to her ear

and screamed. There was a hermit crab inside and it pinched her ear.

She never wants to go back! LoL I know this is completely off topic but I

had to tell someone!

Hahahahaha I always have things I have to share to. I have 4 boys, 14y, 13y, 6y, & 19m, and they are all nuts!! Haha you made me laugh! 😊

I hVe 4 boys as well. Yay. Yup. Always something random to share 😆

Would it be possible to get a bit more info on how to read this pattern? I’m struggling with what the actual numbers mean. For example, does 2, inc, 3, inc mean stitch 1 sc in next 2 chains, then in the next chain, stitch 2 sc into the next stitch, etc…?

I do hope you’re receiving comments on this so many years later- I was hoping for more information on using your math to scale the sphere to much bigger proportions, I was hoping for roughly a basketball-sized sphere. Sadly, it’s been many years since I forgot how to do this kind of math. Thanks!

OMG, I loved how you used math and crochet! Those are like my 2 favorite things in the world! Lol. Thank you so much for this pattern. 😀

Hi! I am currently working up your 30 row sphere with super chunky yarn and an N hook… but I was wondering, is there any way I could get you to write up a pattern for a 80 row sphere? I am attempting to make a giant pouf pillow. The 30 row pattern with chunky yard is turning out to be about 12 inches in diameter, but I’d like to make one that is much bigger.

Thanks so much for sharing. My 12 inch sphere is going to make a fun pillow.

In the pattern, when you say

example: 5, inc, 5 (11) = “Sc in each of next 5 stitches. Inc in next stitch. Sc in each of next 5 stitches. (11 stitches in this row).”

I think that should be 12 stitches after increasing

Love your pattern, thanks,

I’m working on the 22 row sphere and can’t seem to get passed row 8. 6,inc, II, inc, II, inc, 5 (39) Please help.

Hi, Here is the row in regular (non jibbery) language. “Single crochet in the next 6 stitches. 2 single crochet in the next stitch. Single crochet in the next 11 stitches. 2 Single crochet in the next stitch. Single crochet in the next 11 stitches. 2 single crochet in the next stitch. Single crochet in the next 5 stitches”. Hope this helps!

Thank you for sharing, my sphere came out great!

I was just wandering how you calculated the ratio between the amount of rows and the maximum amount of sticthes i.e. 22 stitches near sin(pi/2) for a 10 row sphere. Did you use certain height to width ratio for a sc?

I think I used roughly a 1:1 ratio for a single crochet stitch.

Reading this post sounded like this in my head: ‘Blah, blah, blah, math, blah, blah-de-blah, blah, blah, math, blah.’ I have NO idea what all of the math nonsense means, but I can’t wait to try your pattern! Thanks for figuring it out for those of us who don’t do math!

Haha that’s exactly what I heard. We were paying attention in class. 😉😁

Probably common crochet sence, but these are worked in a spiral, right? No joining? TIA

Yes, that’s correct!

Does inc mean increase in the pdf? I am fairly new to crocheting and don’t understand all the lingo yet so if you could give me something like a table of your abbreviations I would be very appreciative. 🙂

Yup – you’ve got it. The explanations for the abbreviations will be in the pdf near the top.

Thanks a bunch!!!! I’m gonna give these a go right away!!!

Just started making amigurumi’s and I did not like the systematical decs that give slight ribs…was about to rethink those…but you’ve done it for us!! Many thanks from a mathematical crocheteer ;-))

Can I Marry you? Ok… Maybe not, but you are officially on my idol list! Math AND crochet, Get OUTTA HERE with that stuff!! I have now pinned you and will be scouring your blog for other useful tidbits! Thank you!

Much Respect! RESPECT!!!

Kitty

Absolutely fantastic. Thank you so much for figuring this out and sharing with the world.

Fabulous, thanks for your work. I used your tuto and the result is perfect. I would ask you to authorize me to traduce it in french in my blog, for many of my friends don’t speak english and would adore to use it.

Great mathematics.. I couldn’t read the labels on the axes… print was too tiny on my screen, so it took me a little longer than it should have to follow what you did. Hope you can add larger labels. I’ll be bookmarking this site.

Is it weird to say that I loooooove you?

No. I don’t think it is, because I love her too now! Haha

My UC-Berkeley Math major Mom would LOVE this!! I need to make one in her memory.

Thank you! It’s very interesting and useful!

Thanks for the sphere!

0.o I’m so confused, but Thank You, you’re awesome.

If we create a pattern based on your tutorial, is it ok to sell the pattern if we include acknowledgement?

I created an amigurumi pattern using your tutorial and I just want to make sure what is acceptable, before I decide to sell the pattern.

Hello, Anything that I offer on my site is just for people to use for fun and not to sell as their own. Please do not use my work in a pattern that you wish to sell. I offer this as something for people to make for fun and for free.

Thank you,

Emily

Thank you for the reply. I figured as much but wanted to make sure!

Reblogged this on Everything Anna and commented:

Testing this baby out!

What are the finished dimensions of the balls? I’m looking for a 12 inch or so size. Thank you

Thanks very much for sharing this, it’s great and I would never be able to figure this out on my own.

http://freecrochetpattern.weebly.com/

Доброго вечора! Дякую за вашу статтю і за те, що поділились своїми розрахунками.

I found the link to your site while reading Moogly. Thanks so much for sharing this. It was very easy to follow. :

That is awesome! Um…I don’t suppose you could do an ideal flat circle?

This pattern is PERFECT!!! Thank YOU. I am also interested in a ideal flat circle. My Dr. Who Fez top never turns out flat enough for me.

Hi

My name is Marianne.

I am starting up a forum for crochet women in Denmark, we are only just finding our feet and are working out how to get started.

We have a Links Library, everything to do with crochet – only small so far, but it’s a start 🙂

There is now a link to you: https://mspremiseconclusion.wordpress.com/2010/03/14/the-ideal-crochet-sphere/

We have taken the liberty using a single photo from the DIY/pattern to represent the link, so that it is easy to see what it is.

You can access our Links Library here: http://www.haeklenaalen.dk/links

It is open for guests.

We might also, in the furture want to make other links to you, using other pictures, to show with the links.

If you do not want us to use your pictures with links to you, please let us know, we will then remove them right away.

You can contact me at this email address: kontor@haeklenaalen.dk

Love from Denmark

– Marianne

http://www.haeklenaalen.dk

Many thanks for sharing this with the people of yarn ^_^

Hi! I am wondering if you can tell me about what size each row equals out to? I am looking forward to making this, however, not sure what “row” size to stop at. A toddler size is needed. Thank you.

Alicia

Maybe needing about 16″-19″ if possible. Thanks. (See previous comment). Thank you.

Hi Alicia, If you’re looking for a sphere that large, you might want to check out the customizable sphere post – you just need to input how large you’d like your sphere to be and it will generate the pattern for you!

https://mspremiseconclusion.wordpress.com/2014/03/21/inspiration-and-a-personalized-sphere/

Thank you thank you thank you!!!! I’m currently keeping two kids who have a lot of issues and the younger sister loves anything spongey. After her biting holes in a nerf ball I decided I’d crochet a couple of balls for them to play with. This is PERFECT!!!!

Great info. Lucky me I recently found your website by chance (stumbleupon).

I have saved as a favorite for later!

whoah this weblog is excellent i like studying your posts.Keep up thee

great work! You understand, many individuals are searching round

for thijs information, you cann help them greatly.

Thanks for sharing your info. I really appreciate your efforts and I

will be waiting for your further post thanks once again.

I love these! Thanks for sharing!

For the 10 row sphere it reads:

Row 10) 4 scdec.

Could you please explain this? I’m pretty new to crochet, and cannot figure this last row out?

Thanks!

Hi Sally, I’m glad you’re enjoying the sphere! The final row is just 4 single crochet decreases in a row. You won’t complete the entire row, but you should fasten off after this point. This is so that the base of the sphere doesn’t look pointy.

Cheers!

Emily

I can’t get to the pdf file either. I would love to try this 😕

The PDF link is just under the last sphere photo. See where it says PDF, it should be a different color. Click on it but you need to have some type of PDF support software like Adobe. Just search pdf software in Google. You should pay nothing for this. If you do this or already have it then it’s probably a matter of finding where the pattern file was saved. For this you can do a search for the file name. If you need anymore help find me on Facebook or reply to this, I will try my best to help.

This is awesome. Thanks!

Are you the best… or are you the VERY best!? Thanks so much for this! I love the math. I love the fiber-craft. I love the Simpsons reference. You’ve put a smile on my face! 🙂

Hello! I do not understand what the formula. Or is it secret?

Glad to be one of the numerous visitors of this excellent site.

Wow! This is super helpful! Thank you for this! (I’m also really impressed at you application of math!)

I’ve been frustrated with my favorite amigurumi pattern book because I could never get their heads to come out right. I’d been looking for a new head pattern and you just solved my problem in an awesome way.

Awesome work!

Can someone tell me what size ball each pattern makes? I want to make a 1″ ball…think Cake Pops! And I don;t know which pattern to choose.

They work up fairly quickly and as you get to about the 3-5 row you will clearly start to see the size. That said it has been a while since I made one but I believe I used a 3.75mm hook and made the 10 or 12 row ball and it came out a perfect 1″, you can see the results here http://www.ravelry.com/projects/irishhoney/weeping-angel-statue. I suggest you just start working up a sample. Hope I helped, I would love to see a pic of your finished project and let us all know what worked for you.

Have fun.

Question: I have a sphere that is 33 rows and has 96 stitches at the most so how would I figure out the increases for rows? I am in awe of math and think it’s great but I have no idea how to use it!

Thank you for any help you can give me!

Is this done in a spiral?

うん！最後に|％キーワード1％程度何か誰かが書き込み。

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この上 |スパム問題問題を持っています。私もブロガーだ、と私はあった好奇心あなたの状況を、 私たちの多く 開発しているいくつかの素晴らしい手続きと我々は見ているの貿易と戦略 他の人々興味があれば| 電子メール、電子メール、私は撃つください。

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春新作 即日出荷 http://krontech.no

Thanks so much! Math rules! 🙂

I love this pattern. What I’ve been looking for, for a long time.

Hi, I apologize if you already mentioned this…can you share what size hook and weight yarn you use? Can’t wait to try it! Looks beautiful.

Hi there, I use a 3.75 mm hook with worsted weight yarn. But really, as long as you can make stitches tight enough so that the stuffing won’t poke through, you can use whatever yarn/hook combination that you like!

Hi There,

I am not sure if you are still checking this link. But I wonder if you could help. Your somewhere are awesome. So clever.

I made your 30 row sphere for a project, but it wasn’t big enough for my project. I need something slightly bigger. Could you tell me please if you don’t mind, how to increase it to 35 or 40 rows.

I would appreciate the help. Thank you so much.

Louisa

18 row sphere, row 17. If you start with 17 sts & (dec,1) around then you will be 1 st short bc there’s no lady st to put the 1 sc into. You will end up with 11 sts, last st being a dec. I went ahead & did alt (dec,1) and (dec,2) around like the 16 row sphere to get 12.

My name is Laura i need

Help i dont know what to do on this line

Row 3) 1, inc, 2, inc, 2, inc, 2, inc, 1 (16)

I know it is somthijng easy and i am going to say duh

So, all the stitches are single crochets… so 1=”1 single crochet”, 2=”2 single crochets. And inc=”increase”, which would mean 2 sc in the next st. Hope that helps!

Thank you i was making it hard when it was not

Duh i thing i got it

Yоu can play it anytime you want.? Daddy answered.

?As a result of speaking about how nice God is makes him colmpletely happy and

іts worship. Play itt before you go to slｅep tonight and onc yoս get ᥙp in the morning and God cɑn be close to you all day lօng.

VIANA, Dirce Laplaca; PETENUSSO, Marcio. http://cgi.members.interq.or.jp/blue/mayumi/fantasy/fantasy.cgi?&details=594

Thank you so much for sharing this. I work with kids and testing can get a bit stressful for them. These spheres have allowed me to make them some fun stressballs to mess with while they are working.

Thank you for your generosity in sharing this. What fun using math skills. Was “shrinking” a pattern the other day using proper proportions and percentages. Very satisfying to use math in crochet. Regards, K

I really love your sphere pattern, thank you so much for sharing it!

On trying the 24 row sphere I am surprised that row 10 increases by 3 stitches whereas the prior row increases by 2 stitches only. Shouldn’t the rows be switched?

Thanks for this pattern – I’m making spheres in different sizes for my kid. Having some trouble figuring out how to end, though. What does 4 scdc mean?

Hi there, I suggest doing just 4 single crochet decrease stitches on the final row and then fastening off. I find if you do 6 scdec stitches on the last row, the bottom ends up looking too pointy. 🙂

Obrigado pela esfera! Math rocks e você também!😀

hi!,I like your writing very a lot! share we be in contact more about your post on AOL?

I require an expert on this house to solve my problem. May be that is you!

Looking forward to peer you.

This takes a lot more counting and paying attention than other sphere patterns I have used, but it is great! I started the pattern with a magic circle because it closes tighter imo.

I wish all the ‘mathy’ bit was on the PDF! It’s so fun! I grew up with an engineer dad and brother who got his degree in physics, and my son is now in engineering school too. So while my degree is in English, and I often don’t get their math conversations, I appreciate them, and enjoyed reading your posts so very much.

Thank you for the PDF file. Will try soon! 😘

This looks so great. My SO is currently trying to crochet a heart with hemispheres based on your calculations, but none of the sizes quite fit. I can’t tell if I’m missing something because I don’t crochet, or if I’m just failing at math, but I can’t figure out how you calculate the stitches per row. Do you need to start with your desired row count, or desired circumference, or both…? Could you please explain your formula for calculating the correct stitches per row?

Hi There, Sorry for the delay in getting back to you. I hope the heart pattern is going well – it sounds so cool! When I calculate the stitch count for anything, I use the circumference and the approximate size of a single stitch. So, if I know the radius of the row (assuming that row is circular in shape), I would calculate the circumference as C = Pi*r^2. Then, I would calculate the number of stitches (N) by dividing the circumference (C) by the size of a single stitch (s): N = C/s.

Cheers!

Emily

That makes perfect sense, thanks. But I think I phrased my question poorly. My question is not so much on calculating the stitch count for a row of known radius (or vice versa), but rather on the relationship between row count and radius of the sphere. That is, given a radius R of the desired sphere, how do you calculate the number of rows required to make a sphere with that radius? Or, conversely, given a count N of the desired number of rows, how do you calculate the radius (or, equivalently, stitch count) of a sphere with that many rows? I don’t know if it’s easily calculable in both directions, but either way would be very helpful. I think I understand how to get the radius of a given row once the sphere’s radius and row count are both known, but not how to get from just one of those pieces to the whole picture.

Sorry for the miscommunication, and thanks a bunch!