How can I calculate all possible combinations between a specific ratio range?

Question

Not sure this is the right wording for this sort of problem, but let me explain by context.

Instagram displays specific image ratios when posted to a feed. These ratios are:

1:1 (square)

1.91:1 (landscape/horizontal)

4:5 (portrait/vertical)

On the company's help forum, they also state:

[...] the photo's aspect ratio is between 1.91:1 and 4:5 [...]

This specific wording makes it seem like there are other acceptable image proportions in their platform. What I want to do is test this hypothesis by posting other aspect rations that lie between 1.91:1 and 4:5. I know for a fact that 3:2-sized images works.

Is there a formula, trick or correct way to calculate or generate all possible whole values between these two ratios?

Answer 1

You can produce the desired sequence of ratios using a Farey sequence:

the Farey sequence of order $n$ is the sequence of completely reduced fractions, either between 0 and 1, or without this restriction, which when in lowest terms have denominators less than or equal to $n$, arranged in order of increasing size.

That article shows a simple algorithm to produce the next term of the Farey sequence of order $n$ from the previous two terms. That algorithm starts at $0/1$, but we can adapt it to start from $4/5$.

One important property of any Farey sequence is that given two adjacent terms $a/b, c/d$ (with $a/b < c/d$), then $c/d - a/b = 1 / bd$. We can use that to find the next term after $4/5$.

Let $a/b = 4/5$. Thus $$c/d - 4/5 = 1/5d$$ $$5c - 4d = 1$$

Now, for any $t$, $$5(1+4t) - 4(1+5t) = 1$$ So we can set $$c=1+4t$$ $$d=1+5t$$ for some suitable $t$. It can be shown that the resulting $c, d$ are coprime, so $c/d$ is a fraction in lowest terms.

We choose $t$ such that the resulting $d$ is as large as possible with $d\le n$. That is, $$d = 1+5t \le n$$ So $$t=\lfloor(n-1)/5\rfloor$$

---

Here's some Python code (derived from the Wikipedia example) which implements this algorithm. It uses a Sage feature to read the input parameter $n$, but it can be easily adapted to use the standard Python procedures for getting input data.

# Partial Farey sequence. PM 2Ring 2021.09.15
# See https://math.stackexchange.com/q/4250558/207316
@interact
def main(n=12):
    n = int(n)

    # Minimum aspect ratio
    a, b = 4, 5

    # Second term
    t = (n - 1) // 5
    c, d = 4 * t + 1, 5 * t + 1

    while True:
        r = c / d
        print(f"{c:2} / {d:2} = {r:.4}")
        if r >= 1.91:
            break
        # Next term
        k = (n + b) // d
        a, b, c, d = c, d, k * c - a, k * d - b

Here's the output for $n=12$:

 9 / 11 = 0.8182
 5 /  6 = 0.8333
 6 /  7 = 0.8571
 7 /  8 = 0.875
 8 /  9 = 0.8889
 9 / 10 = 0.9
10 / 11 = 0.9091
11 / 12 = 0.9167
 1 /  1 = 1.0
13 / 12 = 1.083
12 / 11 = 1.091
11 / 10 = 1.1
10 /  9 = 1.111
 9 /  8 = 1.125
 8 /  7 = 1.143
 7 /  6 = 1.167
13 / 11 = 1.182
 6 /  5 = 1.2
11 /  9 = 1.222
 5 /  4 = 1.25
14 / 11 = 1.273
 9 /  7 = 1.286
13 / 10 = 1.3
 4 /  3 = 1.333
15 / 11 = 1.364
11 /  8 = 1.375
 7 /  5 = 1.4
17 / 12 = 1.417
10 /  7 = 1.429
13 /  9 = 1.444
16 / 11 = 1.455
 3 /  2 = 1.5
17 / 11 = 1.545
14 /  9 = 1.556
11 /  7 = 1.571
19 / 12 = 1.583
 8 /  5 = 1.6
13 /  8 = 1.625
18 / 11 = 1.636
 5 /  3 = 1.667
17 / 10 = 1.7
12 /  7 = 1.714
19 / 11 = 1.727
 7 /  4 = 1.75
16 /  9 = 1.778
 9 /  5 = 1.8
20 / 11 = 1.818
11 /  6 = 1.833
13 /  7 = 1.857
15 /  8 = 1.875
17 /  9 = 1.889
19 / 10 = 1.9
21 / 11 = 1.909
23 / 12 = 1.917

And here's a link to a live version of the script, which runs on the SageMathCell server.