## Introduction: Hand-Drawn Voronoi Diagrams

If you are into modern art, architecture, digital fabrication, or even geography then there is a good chance that you have stumbled across something called a Voronoi diagram. These honeycomb-like, asymmetric, mesh shapes are used in many types of mathematical analyses as well as to create interesting fill patterns in things like furniture and wall panels. In this quick Instructable I will show you how to create these unique diagrams by hand. That's right. Just you, some paper, a ruler, and some sharp pencils. The old-fashioned way.

Generating these patterns takes seconds on the computer, so drawing one manually may seem a bit ridiculous. However, I can guarantee you that drawing one by hand is the best possible way to learn how these ingenious little guys work. It is also a relatively simple, fun, and relaxing process that is perfect for a rainy day or a long, boring meeting.

As with most of my Instructables, I like to start out with a brief lesson. The subject today is history (and some math, of course). Let's get started.

## Step 1: Russian Math Bros

Georgy Voronoy was a Russian mathematician. The diagram that bears his name is used to divide a plane filled with unique nodes into separate regions. The cool thing about these regions is that at any point within them, you are closer to the node they contain than any other node, and, at any point along their boundaries, you are equidistant to at least two nodes. This makes them very useful for many applications such as mapping and zoning.

Boris Delaunay, another Russian mathematician and a student of Voronoy's, developed a method for connecting the same nodes into triangular regions, which is essential in the process of creating Voronoi diagrams. The key thing in a Delaunay triangulation is that, in each triangle generated, no other nodes exist within the circumcircle of that particular triangle. This is a fancy way of saying that each triangle is formed by connecting each node to its nearest neighbors.

Oddly enough, Delaunay was also an accomplished mountain climber, which may explain why his triangulation method is often used to build the TIN surfaces used to model terrain in 3D. We'll save that for another day...

For those of you wondering about that goofy circumcircle...it is a circle that intersects each of a triangle's vertices. Its center is located where the perpendicular bisectors of each of the three sides intersect, and its radius is the distance from this point to any one of the three vertices. Although we won't necessarily have to draw all the circumcircles later, the center points just mentioned will be very important. More on that as it comes.

So, thanks to these two cool dudes, we can perform some pretty awesome analyses and make some interesting looking artwork as well.

If you would like to read up a little more then here are some links:

Delaunay Triangulation: https://en.wikipedia.org/wiki/Delaunay_triangulat...

Voronoi Pattern: https://en.wikipedia.org/wiki/Voronoi_diagram

Boris Delaunay: https://en.wikipedia.org/wiki/Boris_Delaunay

Georgy Voronoy: https://en.wikipedia.org/wiki/Georgy_Voronoy

## Step 2: Materials Needed

Tools and materials needed to produce a basic sketch are very minimal. Using these as a guideline you can easily scale up to a larger format or medium.

- Pen(s) or Pencil(s) - 4 colors recommended
- Paper
- Ruler
- Square or Protractor
- Compass (optional)
- Thick marker (optional)

## Step 3: Loads of Nodes

We'll begin by drawing some dots. The only important guideline is that you place them as randomly as possible. Maximum and minimum point spacing should be determined based on your individual goals. Tighter spacing will produce a smaller shapes in the final pattern. Conversely, loose spacing will produce larger shapes. More dots = more complexity = more time. Your call.

Your nodes don't have to be a particular size so long as you can distinguish them later.

PRO-TIP: Having difficulty with the randomness? Sprinkle a pinch of any dry, granular, dark material (pepper, sand, sprinkles, etc.) on top of you paper and place a node wherever the pieces fall. You could also spell out a name, trace points on a map, or any number of other things.

## Step 4: Delaunay Triangulation

This may be the most difficult part of the entire process. As was mentioned previously, we need to connect each node to its nearest neighbors, forming a network of triangles.

Another way to describe "nearest neighbors" is that, at a given node, we want to connect it to the the two adjacent nodes that make up a triangle with the smallest area possible. To do otherwise would mean that other nodes will fall within our circumcircle, which is wrong. With most points, finding the nearest neighbors is very intuitive. For those that are not so obvious, use your ruler to compare distances between other nodes in question.

As you begin to connect nodes and form triangles things will start getting easier. You will likely encounter scenarios where there are no other options other than to connect two nodes and complete an already partially formed triangle. Just be sure that you don't leave any non-triangular shapes between your nodes and __don't ever cross another line__ (triangles can't share spaces).

I think the Delaunay diagram looks fairly cool on its own, but let's keep going. We don't want our old pal Georgy to get jealous.

## Step 5: Stuck in the Middle

Once all of the nodes are connected, our next task is to draw the perpendicular bisectors for each line. The obvious first step in achieving this is to divide each line in half. Do this using the ruler and a different color pen/pencil.

Measuring each line and calculating the midpoint, one-by-one, makes this a somewhat tedious step. For this reason, I used a small tickmark to mark all of the midpoints before proceeding to the next step.

PRO-TIP: User nax left a great tip for finding the perpendicular bisectors using a compass. Basically, what you do is place the point of your compass on one of the the nodes, extend it roughly past the middle of the segment, and draw an arc above and below the line. Then you repeat the same process for the point on the opposite end of the segment. The perpendicular bisector will pass through the points where the arcs intersect. I think this is a superior method than my own, but you can decide for yourself. You can read more about it in the comments below. Thanks nax for pointing this out!

## Step 6: Bisector Inspector

Using the square and the same color you used to mark the midpoints, draw a line through each midpoint and perpendicular to the line it bisects. These are the lines that will form the boundaries of the Voronoi pattern.

This has the potential to get sloppy really fast, so take your time and be aware that the perpendicular bisection lines for each side of each triangle will intersect at a single point. Recall, this single point is the center of the circumcircle for that triangle. Neat, huh?

Optional Step: If you still don't believe me about this whole circumcircle thing, or would just like to check your work thus far, grab a compass and test it out. A circle drawn with its center at the intersection of the three lines you just drew and a radius of the distance to one of the points in the triangle should intersect the other two points in the triangle perfectly.

## Step 7: Connect the Dots (again)

It is finally time to reveal the Voronoi diagram! To do so all we need to do is start connecting each of the points where the three bisectors intersect (crazy cool circumcircle centerpoints). The lines you draw to connect them will follow the paths of the perpendicular bisector lines you drew in the last step.

I find it easiest to start on a prominent, central node where the bounding region is well-defined and intuitive.

A fourth color pen or a thicker line will help bring out your pattern against the busy background. However, a thick-tipped marker works the best.

And with that you are DONE! Way to go! Georgy would be so proud!

## Step 8: Roll Your Own

Besides these basic sketches, there are many, many applications for Voronoi diagrams (and Delaunay triangulations too!). Pictured above are a few of my own creations.

Mastery of this technique will allow those of you without large format CNC machines or digital design tools to create and incorporate these patterns into your own handmade pieces. Please post a picture below in the comments of any Voronoi art, maps, etc. that you create.

## Step 9: Wrapping It Up

I hope that you had fun learning about these cool diagrams and even more fun drawing them. Other than some sweet looking sketches I hope I have left you with a solid understanding and appreciation for how they come to be. Now not only will you be able to recognize them anywhere, but you will be able to explain to those lesser informed about how they work and how easy they are to create.

I use each of these patterns in a lot of other work I do, so chances are you will see them again in a future Instructable. Please follow me so you don't miss out.

Thanks again for reading!

I used your instructable to make this painting for my apartment! I'm pretty proud of it.

You are right to be proud...that looks AWESOME! I really like your color choices and how you framed it. Thanks for sharing! I will definitely be looking to your piece for inspiration as I decorate my home.

Thank You !!

Happy to help!

It´s amazing the way things get together. So cool!

Thanks! They are a lot of fun to make, and it's always exciting because you never really know how they'll turn out.

Best instructions on hand-drawn Voronoi patterns yet! I am quite familiar and have researched the topic, so what I say is (as far as I know) true. Thank you for your documentation.

Nice work and I'm flattered by the endorsement! :D

Like snowflakes, no two are ever the same. Really great 'ible and history to fill in. I totally believe in the manual method first to cement in the concepts, and you did it very well. Thanks for sharing!

Thank you!

Great documentation skills!! Love the choice of music and the project is excellent :D Thanks for sharing.

Thanks! I was worried that the music may be too much for some people, but I'm happy to hear you liked it!

To carry your images into 3 Dimentional art and excite those who dont understand maths ,make any rough shape using straight or crocked lines with wire ,all joining especially around outside and immerse fully in a kids bubble blowing fluid and watch natures maths work everything out for you.making intersecting joining lines and bubbles

Photograph it,make a solid model from your photo ,then everyone will think you a mathematical,architectual Genius, Just for Fun

great project, arts and maths together

Thanks! They go together much better than most people think.

Here's my attempt! I hope I didn't botch it. Thanks again for sharing this project. It was the first time I had a great time doing maths :p

You did great!

Happy to share and glad you enjoyed it. Math can be lots of fun!

I'm making a project to enter to an art school and it has been difficult to find inspiration to make 12 drawings.... When I saw your project I saw the light :p Sure I will use it to make one design :)

By the way, I really hate maths... in fact, I suck on maths. But you manage to get me into doing some maths :p

Thanks for sharing it! This is so cool and the explanations very clear.

look into compass and straight edge geometry. you'd be suprised what kind of stuff you can do without a ruler or protractor....

Neat! I made the Voronoi Diagrams with a compass and it was easy than expected. Thanks for the tip :)

I read somewhere - in patterns those who are "artists" see art, those who are "mathematicians" see math. I lean to artists and wish I was stellar at math but am not. Comfort in knowing however one views the subject it is the same!

You're not the only one, I assure you :) Other people I know (the majority) are the exact opposite. That makes for interesting discussions.

Awesome! Thanks for the kind words and good luck on your project.

this is awesome, need to find a pen and som paper to try it out..

Thanks! Have fun!

Wow! Finally I know how to do it with out computer!

Mike's kinda messy but for a first go I'm pleased with it.

You can see the downside to using a compass. :P if I used a thin sharpie the pencil would be less noticeable but.... XP

Sweet! Thanks for sharing and for your feedback on using the compass. Some clutter seems inevitable no matter which method you use. Maybe with a lot of practice I'll be able to draw them without measuring...

Also appreciate the diagrams you made down below. I ran into that exact same scenario and that's a wonderful way of explaining it.

Since you have the compass out anyway, there is an easier and more accurate way to draw perpendicular bisectors than measuring, dividing, and then trying to get the square in there.

These instructions are a bit long because I'm making them very step-by-step. This isn't as complicated as it might look.

In the picture, the blue segment from A to B (call it AB) is the original that we want to draw a perpendicular bisector for.

Start by putting the compass point on A. In the picture, the compass is set for the distance from A to B, but it doesn't have to be as long as it is more than half of that, and you don't change it between the next two steps.

Now draw a little bit of arc on each side of AB about where you guess it goes. If you don't want to guess, you can just draw the whole circle. After you draw a few of these you'll be able to guess. I drew these arcs black in the picture.

Do the same thing on B.

The circles will cross in two places. The line through those two places (I drew it red in the picture) is the perpendicular bisector.

you've doomed me..... XP. i love making things with the compass. it's bizarrely fun to try and make shapes with just a compass and straight edge. oh well, i was probably going to do this anyway, given the pattern is used in blender XP

That is definitely a better method. I've seen it before, so I can't think of a good reason that I left it out other than using the compass is not typically my first instinct.

Thank you so much for pointing this out. When I get a chance I will add it to the midpoint step. Will be sure to give you the credit too!

Thanks again for your help and for reading.

Nice voronoi diagrams. Here's my first try at one. I made a few mistakes, but I think it still looks cool.

i cant answer it all, but i do know how to fix the bit in the top right in the future. :P

as a 3d modeler, sometimes you have to repair a polygon that's not bending the right way. from this you learn that there's always two ways to connect a set of four points to make two triangles.

when i was doing this project myself (pics soon!) i made the same mistake, but i realised what i had done. the problem lies in how you connected your points, represented by the yellow lines in the picture below ((i forgot the center beam, but you get the idea.))

connect them like they are with the green lines, and suddenly you have a new division. which just so happens to bisect the diamond shape created by the 'wrong' method. :D

so, even if you mess up on the original triangles, once you see that overlap in points, you can cheat and just connect the two crossing points :P

hope this helps!

Nice job! Much better than my first attempt...

The mistakes will disappear as you gain more experience. Thanks for sharing!

the whole set is interesting, but using the compass technique you can actually bypass the step where you connect the triangles, and just draw the bisectors instead.

that is... if you can keep your lines seperated.

Cool! Would be great to make with that a garden road.

Awesome! Thanks for the kind words and good luck on your project.

Cool, thanks for sharing!

No problem. Glad you enjoyed it!

Thank You! i was looking for something like this quite a long time :)

You're welcome! I hadn't seen anything like it either so I thought I would give it a try.

So Great!

Thanks!