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.

is the square or rectangle you are about to split. Red lines are new lines you should draw.

are new lines you should draw. Small red dots on the border show where new lines start or end.

on the show where start or end. Green lines are pre-existing lines that should be used as references.

are pre-existing lines that should be used as references. Yellow circles mark the intersections of the green lines to use.

mark the intersections of the to use. Gray lines are old lines that will not be used.

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!