Counting with your eyes

I invented a new way of counting visually patterned items with your eyes

I invented a new method of counting things with your eyes, "bisection counting" of periodic setructures. The idea is to use your eye to spot the midpoint of a linear/radial group of things (eg: fence palings, wheel spokes, or tiles), then keep halving the group again and again until you're down to just one (or nearest to one) item. Raise 2 to the power of halves you counted and you get a pretty fast and accurate estimate.

Calculate the Total:

Use the formula:

\[\text{Estimated Total} = 2^{\text{Number of Halvings}}\]

So if you halved 4 times:

\[\text{Estimated Total}=2^4=16\]

Error Bound estimation

• You visually estimate \(n\) halvings
• Your total estimate is \(2^n\)
• The actual count is \(N\)

The maximum relative error occurs between powers of two:

If \(N\) is between \(2^n\) and \(2^n+1\), and you guessed \(n\), then:

\[ \text{Relative error} = \frac{|2^n - N|}{N}\]

Which means the maximum error is just under 50% (specifically, ~41% at the midpoint between powers of two).

In practice, the actual error would be much smaller, because:

• Users rarely misjudge an entire halving level.
• The uncertainty is usually confined to the last halving, which only introduces small errors, not a full ±1 in \(N\).

So typical errors would be lower, around 5–15%