Mathematical Law that predicts amount of breath for each level of Heightenings

[Mellosaurio]

 

Reading Warbreaker and the amount of breath for each heightening, I noticed that the amount of breath followed some kind of exponential function. So I tried to find the math law that they follow

I started by plotting the amount of breath versus the level (Figure 1) taking into account an error of 1% for each level. This is because it is written that the amount for each level is not exact, so I assume a value that increases as the level does.

↑Figure 1: BioChromatic Breath for each Heightening

Then I tried different basic functions, but I found that it doesn’t follow those. Eventually I divided every amount by 50 (Figure 2) and by the level squared (Figure 3).

↑Figure 2: Figure 1 divided by 50

↑Figure 3: Figure 2 divided by Level Squared

With this i started to try again with this formula () and found values for A, B and C that approximated the curve. then i expanded the formula so the results approximated Figure 1, i got

Eq. (1)

Using OriginLab I plotted Figure 1, and fitted using Eq. (1). And i got Figure 4:

 

↑Figure 4: Figure 1 fitted using Eq. (1)

The values for this formula are:

Replacing the level we got this values:

 

This aren’t very accurate, but they are very near, and may be the real values that are approximated in the books. Be aware that there is an unknown error to this values, they could be less or more

Conclusion

The most probable thing is that Brandon picked random values for each level, but it still was fun to try and find out this law. 

Replacing for an 11th heightening we get around 120000 breaths for it. If in the future there are more levels, and they follow this formula i would be very happy. Keep in mind they will most probably not follow this.

[Mellosaurio]

I wasn’t completely happy with the previous formula, it was messy and imprecise. So i followed my previous reasoning of dividing everything py power of the level, and with the cube i got a nice little parabolic function (Figure 1), which i fitted using

↑Figure 1: Breath divided by the level cubed

 

Of course, some of the values don’t fit very well, and with the final formula, those are the farthest of the original values.

Then i multiplied the previous formula by x³ and got

And i with this i fitted the original values (Figure 2)

↑Figure 2: BioChromatic Breath vs Heightening level, fitted

 

and with this i got

The predicted breath for each heightening is:

 

And for an 11th: (120623 ± 9.576)

Conclusion

 

I believe this Formula is a LOT better than the last i got, and much more precise.

[Mage he/him]

My fingers itch to pull out my graphing calculator and plug all this in. 

[Hoid the Former Drifter he/him]

shoot! haven't done math in a while.

[Zoey she/her]

Someone should try to ask Brandon about this, and see if he wants to use one of these in the coming Nightblood book. He does have an equation for Awakening planned out, why not for Heightening levels?

 

Some other sets of numbers from it, if you want to try to get anything from them

(breaths / heightening)

50, 100, 200, 250, 400, 583.333, 714.285, 1250, 2222.222, 5000

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((n+1) - (n))

50, 150, 400, 400, 1000, 1500, 1500, 5000, 10000, 10000, 30000

This appears to follow a pattern of 1, 1, 2, 3, 3, 4; 11th Heightening might be 80,000. 

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Average of subsequent Breaths

25, 125, 400, 800, 1500, 2750, 4250, 7500, 15000, 35000

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Each divided by 16

3.125, 12.5, 37.5, 62.5, 125 (neat that the 5th Heightening, associated with Returned, is perfectly divisible by 16), 218.75, 312.5, 625 (very rarely reached, associated with Greater Awakening, also divisible by 16), 1250, 3125 (So all at 8 and above are divisible by 16)

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Divided by 5, sacred on Nalthis

10, 40, 120, 200, 400, 700, 1000, 2000, 4000, 10000

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5th Roots

2.18, 2.88, 3.59, 3.98, 4.57, 5.11, 5.49, 6.30, 7.24, 8.70

 

 

So see if any of these help to get better formulae, or make things more clear. 

 

Edited January 7, 2022 by Zoey

[Wayne's Unlucky Hat he/him]

Dang you are devoted