Divide into equal parts

Below are pattern sequences that can be used to split a square or rectangle into an exact number of equally sized parts without doing any measurements at all. It is accomplished by drawing lines between intersections or corners on the frame or through intersections of the lines inside the frame. Lines can also originate from the horizontal or vertical vanishing points if in perspective, which outside of perspective just means perfectly horizontal or vertical lines.

As I have done some fairly extensive research in this area there are a few patterns that have alternative paths. In these cases the top pattern is either easier to use or less complex while the bottom pattern can be used to actually divide the square or rectangle horizontally as well, creating a grid. All other patterns can already be used to create grids.

The philosophy is that every step adds as many lines as possible which do not depend on each other but previously existing lines. This means that you can draw the new lines in any order you wish.

Document legend for each step:

  • The black frame is the square or rectangle you are about to split.
  • Red lines are new lines you should draw.
  • Small red dots on the border show where new lines start or end.
  • Green lines are pre-existing lines that should be used as references.
  • Yellow circles mark the intersections of the green lines to use.
  • Gray lines are old lines that will not be used.
  • Blue lines are previously drawn lines that are part of the final result, they are faded like the gray lines when not in use as references.

All these patterns have been verified by doing measurements, which again is precisely the activity we are trying to avoid by using them! To easily test a pattern you can use a sheet of paper and fold it according to the instructions, this is of course easier with the earlier patterns that result in fewer parts.

These first four patterns are pretty much all the basic methods used to start more complex patterns. They usually make out step 1-3 in some way, and in reality, that is because no other patterns are possible in those steps! Please click the images to view larger versions!

The first step in this most basic pattern is actually the first step for all the patterns. It simply finds the center of the frame.

It was actually after discovering this pattern when I needed to split something in three that I got interested in exploring geometry in this way. As you can see it is a basic method that can be achieved by drawing lines in several ways.

The top patterns actually starts out just like with thirds above, you just use different intersections to draw the next step. In the bottom step you can see how we repeat the first step om the third step.

This pattern is my favorite by far, so simple yet so useful. A fun thing is that the parts created can be used to make five equally sized squares.

These next five patterns are the ones I have used the most, with the last pattern doing tenths. It is very seldom I want to split something in more parts than that.

The patterns that are the hardest to find are naturally the primes as they cannot be based on any other pattern you already have found, they are all very unique, this is the reason why finding all the patterns in a sequence is a bit tricky.

While I found various patterns for much higher amount of parts I decided to stop at 25 because it would be realistic to fill in the gaps. Even then it is probably a few patterns too many, mostly because as the amount of lines increase the patterns become quite busy and thus hard to read.

I have to confess I spent way too much time figuring all of this out. Hopefully it will be useful for someone else than myself :) If so, do not hesitate to leave a comment!

64 Responses to Divide into equal parts

  1. Akilan says:

    thanks for this separation of boxes i think this make me to draw much quicker without confusion

  2. Thank you :) Glad you have use for it! If you ever drop by again, don’t hesitate to mention what you do, I’m curious!

  3. Yudha Nasution says:

    you’re rock man…, so useful to get learning. Thanks a millions….

  4. topcart says:

    Hi do you have a video that showing an action when you drawing based on all the amazing patterns you’ve been created? Thanks :-) , hope to hear from you soonest

  5. Hi Yudha/topcart! No actually I have not made one yet, I was thinking about it but haven’t decided which format would be best. What would you like to see, the pattern just being drawn on a flat page or in perspective? I cannot promise anything, but it’s nice to get some input :)

  6. Carlos says:

    Thanks so much for sharing those patterns! It’s been a while that I’ve been trying to find a way to divide a rectangle in fifths in perspective, but never found out how. I’ve seen other methods on the web, but yours is by far my favorite.

    It is incredible that you went up to 25!

    Thanks also for sharing your research ;)

  7. I was in the same boat Carlos :D I figured there had to be a simple way, and this is what I ended up doing after a couple of years :) Thank you for your comment!

  8. Peter says:

    Wow! This is very useful for any form of Origami figures! Hopefully one of these prime numbers can be useful to make a humanoid like figure somehow with the leftover flaps.. ;)

  9. Just used the 1/3 for a wood project – thanks

  10. Rob Chansky says:

    I tried to measure with a ruler (for a flag painting) 5 times and it came out wong each time. Thanks for putting this up, I can see it was a lot of effort.

  11. Leon G says:

    You are genius! This is the best answer I found for such questions. Thank you for sharing your great idea with us.

  12. Sofía says:

    Great post!!! Really useful :) And accurate!!! I tried the 1/7 division on a sheet of A4 paper, and the difference between the divisions was less than 1 mm. Thank you so much!!

  13. Happy that you found it useful :) Nice to hear that it worked! :D I’ve validated the patterns in AutoCad, but I guess using the patterns for physical things is completely different, especially when folding material over itself!

  14. Greg says:

    This is so wonderful I’m glad you did this. I’ve been working on some perspective drawings and making the window spacing look right has been difficult. Then after finding this it has worked great.

  15. Gargi Lagu says:

    Thank you very much for the pattern . Very easy to understand

  16. Hi Andrea!

    Since 2013 I have been doing all of my artwork with just a compass and straight edge.

    I create my own grids using this method…however, I’ve NEVER gone above an 8×8 grid.
    This information is priceless!
    Thank you SO much for sharing this knowledge on the web.
    I can only imagine the amount of research….but it’s fun, isn’t it?

    Looking forward to visiting your site in the future.

    Warm Regards.

  17. PS Have you ever read ERIC BROUG’S book titled ISLAMIC GEOMETRIC PATTERNS?
    In it you learn how to divide the square into infinite parts–either diagonally, or vertically and horizontally.
    You can make square grids and or triangular grids.

    I think you would enjoy taking what is taught in this book (and in the enclosed CD-Rom) and simply letting your imagination run wild with infinite possibilities. And if you have any questions, Eric is quite gracious: he will usually reply to questions on his YouTube channel.

    The “Islamic” patterns are, of course, not only indigenous to the Middle East. Many of these patterns were discovered and used by mystics from many different cultures and earlier eras. Since its discovery, the circle has always represented perfection with no beginning or end.

    Peace to you,


  18. abhieet,ankit says:

    thanks for this is our hhw

  19. WodzuWodzu says:

    Excellent work, thank you :)

  20. Dr Amin Firdosi says:

    thanks bhai wonderful !!!!!

  21. To Andreas Aronsson – you wrote so well, and in such lucid & fully detailed construction, that I was distracted away from admiring the drawings & their layout! And I’m an architect, so please accept my comment in that context. . :–)

  22. saiyara shuchi says:

    Can you able to give a short cut answer? That can help us!

  23. Thanks everyone who appreciate this :)

    I’m sorry Saiyara, did you have a question? I don’t see anything to answer!

  24. Yifeng Mu says:

    Awesome work! I’ve been too lazy to figure this out by myself.

  25. michaelflahertydesigns says:

    Thank you so much! I was trying to frame out an arbor with evenly spaced joists and couldn’t quite figure out how to divide the rectangle evenly with a depth that came out to hundredths of an inch (using CAD). This worked like a charm.

  26. Matt says:

    Thanks so much!

  27. Alisa says:

    This was exactly what I needed! I needed this for dividing up window panes for my interior design hw. Lol I was sitting there wondering if all my classmates were division geniuses. So this helped me a lot! Thank you!

  28. Shannon Stenger says:

    Thank you! I have this page bookmarked in my web browser because it has come in handy many times when I use rectangular paper to make origami. (I like the “powers of two” method for dividing squares into nths because it only requires drawing two lines, but it only works for squares.) Very smart, very helpful information – you rock!

  29. Joe Dimanti says:

    Thanks! you have no idea how much simpler it is for me to divide somthing like a square now. Very smart.

  30. Sharon says:

    I hate this! You don’t give the right information that I need

  31. Thank you to everyone who appreciates this guide, and sorry Sharon that this wasn’t helpful for you. It would be of interest to know just what you expected, if you explain this I can consider updating the guide accordingly!

  32. gigi1629 says:

    Thank you!! Now I can incorporate this into my one point and two point linear perspective Art pieces with more accuracy.

  33. Dennis Woods says:

    Thanks, this is absolutely wonderful geometry! You are a lifesaver.

  34. Lisa Olson says:

    Thank’s ! I was figuring something out for a recipe I’m just going to cheat but thanks

  35. Tony says:

    Wow! Profound Stuff! Thanks

  36. Tina Magayam says:

    thanks for the efforts to create all this, It help a lot. appreciate it very much.

  37. Thank you!!! This is great info and really helpful!!

  38. Lee R Vander Baan says:

    So let’s suppose that we impose some thickness to the lines. Is it possible to adapt your drawings to such limitations? I am trying to design an open shelf made from 1/2″ thick wood. The overall dimensions are 14 x 28 and I want to interlock the wood such that there will be 14 smaller rectangles of approximately 1×2. Thanks for this very interesting lesson in geometry (and aesthetics).

  39. Kyuubimode7 says:

    Wow dude you really helped me out :)

  40. Confused says:

    I can see how this works for a square. How does this work for a rectangle? For example, folding a rectangle on both diagonals will not give you four equal parts.

  41. As this technique was designed for perspective drawing, if the root shape is a four point non-twisted one, it will create equal parts but in perspective.

    It works just as well for a flat rectangle though, then you will get equal sized columns, which is a key point. It’s not the preceding folds that get you the equal sizing, it’s the columns. Or grid, if you decide to split in two axes.

  42. MaakuDesu says:

    I just wanted to thank you for putting together this valuable resource!

  43. Zar says:

    Hopefully your efforts will help me cut soap bars from a rectangular mold that I need to divide into 60 pieces of around the same size.

  44. Richard says:

    These work for squares not rectangle as suggested

  45. Richard, I’ve used these methods on practically only rectangles from when I came up with them. Do you have an example of what is not working for you?

  46. Richard says:

    I have what is similar to 8.5 x 11 inch (letter size) rectangle and have been trying to get 3 equal columns (length wise) but can not get an exact match, which I fiqure is impossible, there has to be a way … Thanks for responding, wasn’t expecting a response

  47. Getting back to this, work has been hectic this week so it slipped my mind.

    Splitting in three is one of the more basic patterns that I use most often, and I’ve validated these to work in AutoCAD, with as high precision as I could. It does require accuracy to be completely effective as errors will propagate, but you do make me interested in what is going wrong here.

    If you don’t mind you could contact me through the site form or just post a link to a picture here and I’ll see if I can figure out where things go wrong.

  48. rip234 says:

    how do you do it with rectangular paper

  49. You perform the exact same steps. This is about using corners, intersections and perpendicular lines, references that work with any proportion rectangle.

  50. Pi says:

    Very interesting, thank you very much.

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