Number and Function layer layout

[musjj]

Is there a reason why your number and function layers are laid out like that?

For example, number pads usually look like:

7 8 9
4 5 6
1 2 3

But yours look like:

9 8 7
3 2 1
6 5 4

Is it somehow more ergonomic that way?

[getreuer]

@musjj good question! There is an ergo motivation to this, though to be totally fair, the benefits of doing this are tiny... unless, perhaps, you do data entry or other role that involves typing many numbers.

I was inspired by the post Optimizing the number row, where the author explores nonstandard permutations of the usual number row. I wanted to make a one-handed numpad sort of layout by the same principles.

The motivation for rearranging to something other than the standard digit layout is an observation called Benford's law: lower digits occur more frequently in writing than higher digits. Over a corpus of my own emails, documents, and code, I confirmed Benford's law to hold convincingly. Digits in my writing occur with these frequencies (units are percentages):

Digit	0	1	2	3	4	5	6	7	8	9
%	0.865	0.610	0.514	0.318	0.288	0.259	0.263	0.207	0.264	0.236

I type a lot more 0s, 1s, and 2s than the higher digits. So, like how optimized alpha layouts put the most common letters on home row, this suggests the lower digits ought to have prioritized, comfortable positions. This motivated me to take the standard numpad and (1) swap the bottom and middle rows, and (2) mirroring the layout, since I'm putting the numpad on my left hand:

7 8 9        7 8 9        9 8 7
4 5 6   ->   1 2 3   ->   3 2 1
1 2 3        4 5 6        6 5 4

And I put 0 on a thumb key on my other hand. This has the desired effect of putting the more-used lower digits in nicer positions. IMO, this layout is still close enough to standard that it's also easy to remember, a good balance between layout optimization tweaking and practical simplicity. My left-handed numpad layout:

9 8 7
3 2 1
6 5 4
           0 (right thumb)

Going further...

Alpha layout design puts a lot of attention on minimizing same-finger bigrams (SFBs). So I thought about that too, considering frequencies of typing pairs of digits ("digit bigrams"). Here are the digit bigram stats over my corpus. The row is the first digit and the column is the second digit. Certain pairs are conspicuously more frequent, including 10 and 00 (powers of ten), 16 32 64 (powers of two), and 19 and 20 (years 19xx–20xx).

	0	1	2	3	4	5	6	7	8	9
0	0.047	0.025	0.012	0.003	0.003	0.005	0.007	0.003	0.004	0.003
1	0.023	0.009	0.013	0.004	0.006	0.009	0.038	0.007	0.009	0.021
2	0.052	0.005	0.004	0.003	0.009	0.013	0.002	0.003	0.003	0.001
3	0.006	0.002	0.035	0.001	0.002	0.002	0.002	0.002	0.002	0.001
4	0.005	0.003	0.001	0.001	0.005	0.002	0.001	0.001	0.007	0.002
5	0.009	0.002	0.001	0.001	0.001	0.003	0.003	0.001	0.001	0.001
6	0.005	0.001	0.001	0.002	0.013	0.002	0.002	0.001	0.001	0.001
7	0.003	0.001	0.001	0.001	0.001	0.001	0.002	0.001	0.001	0.001
8	0.008	0.001	0.002	0.001	0.002	0.001	0.001	0.001	0.001	0.001
9	0.004	0.002	0.004	0.003	0.002	0.002	0.006	0.004	0.004	0.008

Based on that, the standard numpad performs with 0.208% SFBs, supposing 0 is pressed with the index finger:

  7 8 9   SFBs = 0.208%
  4 5 6
  1 2 3
  0

Moving 0 to a thumb key improves to 0.114% SFBs:

  7 8 9   SFBs = 0.114%
  4 5 6
  1 2 3
0

A computer-assisted search found this scrambled order to be an improvement yet:

  8 9 4   SFBs = 0.085%
  1 2 3
  7 6 5
0

As an experiment, I tried (a mirrored, left-handed version of) the above scrambled numpad for a while. I was able to learn it, but I always felt some added cognitive overhead in using it, and I didn't feel a noticeable improvement in typing comfort to justify the scrambled order.

My takeaway is an impression that a "light" optimization of the number row / numpad can be helpful to put more used digits in more comfortable places. I mean "light" in the sense that the layout can be post-hoc rationalized as a conceptually simple digit arrangement, one way or another. Going further than that to arbitrary scrambled orders probably isn't worthwhile.

In any case, it has been an interesting experiment! =)