Compromise Dot Paper
3 minutes read

I like graph paper. I always have. But none of the stuff out there quite addresses all my needs.

First off, I like a dot grid more than a line grid. Thankfully, there are lots of really nice dot grid notebooks out there these days. More pernicious, however, is the fact that none of the notebooks I can find work well for every situation. My particular desires may be a bit unusual, but: I want a grid paper that I can write paragraphs on, that I can draw squares/graphs on, that I can draw hexes on easily, and that I can do 3-D sketches of furniture and such. The hex need is primarily for board game design efforts, which I acknowledge is a niche market.

Square Grid Paper

Square paper is great for drawing boxes. You can draw all kinds of great squares and rectangles with a square grid. It’s also pretty great for tabulating and graphing data, and due to its nice straight horizontal lines/dots it works pretty well for longform writing, too.

Unfortunately, if you try to draw a hexagon on square paper, you get a hex that is a bit wonky:

Square Grid

Isometric Grid Paper

Isometric grid paper is composed of a grid of equilateral triangles, and it solves some of the issues with a square grid. Perfect hexes and cool triangles. Depending which way the grid is oriented, it’s alright for writing, too.

Iso Grid

Unfortunately, it just can’t do squares. It’s also bad for making graphs.

Honorable Mention: “3-D Drawing” Grid

Another type of grid I quite like is the 3-D drawing grid. This is really a subgenre of the isometric grid, where the vertical spacing has been smooshed a bit so you can make cool 3-D drawings:

Halfiso Grid

It has basically all the drawbacks of equilateral iso grid paper, plus it makes it a bit harder to do hexagons or equilateral triangles.

Compromise Paper

Neither square grids nor isometric grids are perfect, so I wanted something that combined the best elements of both. Unfortunately, geometry is against us. Squares can be split up into two triangles (sides $1:1:\sqrt(2)$, or four triangles (sides $1:2:\sqrt(5)$), neither of which are the equilateral triangles you need for perfect hexes.

sq2hex2 sq2hex1

Similarly, if you work from the other direction, you start with equilateral triangles. These can be split up into 30-60-90 triangles with side lengths of $1:\sqrt(3):2$. And thus, squares that aren’t quite:


My solution is to split the difference with a grid that does only slightly wonky hexes and only slightly wonky boxes.

I call it the Compromise Grid.

Halfway Happy

Squares need a square grid with a vertical-to-horizontal ratio of 1:1, and perfect hexes can be done on a square grid with a ratio of $\frac{\sqrt(3)}{2}:1$. So I take the middle point, and my ratio becomes $\frac{\sqrt(3)+2}{4}:1$ or roughly 0.933-to-1. Actually, I double up the dot frequency across the horiontal, so the ratio is $\frac{sqrt(3)+2}{2}:1$.


Squares look better and hexes look better. Neither is perfect, thank to geometry, but both are acceptable. In practice, my penmanship is imperfect enough that the variation in my hand’s tolerances exceed that of the grid. I’ve scaled everything so that the vertical spacing is equivalent to quad rule/narrow rule paper. Because it’s straight lines across, it works well for writing. I’ve added darker dots every fourth dot along the horizontal, and every second along the vertical.

As a bonus, if you rotate the paper, you get a pretty workable 3-D drawing grid:


If you’d like to try it out, I’ve made an 8.5x11 version available for download:

Download PDF

Now I just need to get someone from Moleskine or Leuchterm to start making nice notebooks with this dot spacing!

tags:  graph paper  dot paper  square grid  isometric  hexes  boardgame design