# A Teraflop Trajectory: Arrow’s Hidden World of Nanotechnology, Calculus, and Population Dynamics

## Appendix

Calculating max processor speed in flops:

Intel Core i7-4770K at max clock speed of 3.9 GigaHertz.  Hertz is a measure of clock speed, specifically the number of cycles per second.  There is no 100% accurate way to translate between Hertz and flops.  However, there are some approximations.  The modern microprocessor is able to do about 4 floating point operations per cycle.  This processor has 4 cores, but Intel’s hyperthreading doubles the number of cores, so we get 8 cores.  Using the formula cores * clock speed * FLOPs per cycle, we get an overall processing speed of 124,800,000,000 FLOPS per second or 124.8 gigaFLOPS

Unrestricted Growth Formula:

Base formulas come from Techniques for Wildlife Investigations and Management by Clait Braun.

Nt = N0ert where Nt is the population size at time t, N0 is the population at time 0, and r is the rate of population growth.  I am assuming that our starting population is 2 nanobots and that the time was 9 hours.

169 = 2e9r

ln 169 = ln 2e9r

ln 169 = ln 2 + ln e9r

ln 169 = ln 2 + 9r ln e

ln169 – ln 2 = 9r

ln169/2 = 9r

1/9*ln84.5 = r

r = 0.49297

Our new equation is specific to our hypothetical population of nanobots.

Nt = 2e0.49297t

Density Dependent Growth Formula:

Nt = K/(1+ea-rt) where K is the carrying capacity, r is the maximum growth rate if the population were free of constraints and a is the size of the population at time 0 relative to the maximum population size.  I am assuming a population carrying capacity of 200, because I think 200 is a nice number.  We have to make so many assumptions with this data set.  I wish Felicity would answer some hard questions about her nanobots.

169 = 200/(1+e2-9*r)

169(1+e2-9*r) = 200

1+e2-9*r = 200/169

e2-9*r = 1.1834 – 1

ln e2-9*r = ln 0.1834

2 – 9r = -1.6959

-9r = -3.6959

r = 0.41066

Nt = 200/(1+e2-0.41066t)

Integral of 0.006*( 200/(1+e2-0.41066t))

integral 1.2/(e(2-0.41066 x)+1) dx

=  1.2 integral 1/(e(2-0.41066 x)+1) dx

=  -2.92213 integral 1/(eu+1) du

=  -2.92213 integral 1/(s (s+1)) ds

=  -2.92213 integral (1/s-1/(s+1)) ds

=  2.92213 integral 1/(s+1) ds-2.92213 integral 1/s ds

=  2.92213 integral 1/p dp-2.92213 integral 1/s ds

=  2.92213 log(p)-2.92213 integral 1/s ds

=  2.92213 log(p)-2.92213 log(s)+constant

=  2.92213 log(s+1)-2.92213 log(s)+constant

=  2.92213 log(eu+1)-2.92213 log(eu)+constant

=  2.92213 log(e(2-0.41066 x)+1)-2.92213 log(e(2-0.41066 x))+constant

=  2.92213 log(e(0.41066 x-2)+1)+constant

|   =  2.92213 log(7.38906+e(0.41066 x))+constant

Plugging in 9 – 0 for x we get 817.3156

### 5 Comments on “A Teraflop Trajectory: Arrow’s Hidden World of Nanotechnology, Calculus, and Population Dynamics”

1. Dimwit #

The problem is what medium are the nanobots in? It would have to be a Cronenbergian/del Toro semi organic tech to work. A Dell workstation just doesn’t cut it. I don’t see that sort design aesthetic working in Arrow.
Does take the term of “expandability” to new heights though!

2. Shana Mlawski OTI Staff #

Yep. The math checks out. (Says the English teacher with no idea what she’s talking about.)

I will say that I recently binge-watched Arrow, too, and the second season is significantly better than the first, I think because it really owns its ridiculous comic bookishness. But you might want to avoid the episode where Felicity cyber-fights the Clock King because she’s worried she’s not good enough to be on the Arrow team.

3. Steven L. #

“Just a heads up: normal computers don’t work like that. If we took the first derivative of a normal processor’s speed, our answer would be 0 because the derivative of any constant number is 0.”

Except that isn’t true. It’s pretty common these days for CPUs to throttle up and down the processing speed for power/thermal reasons. Furthermore, some chips can be overclocked by the end user to run faster than the officially listed maximum speed by the end user.

4. Jamas Enright #

Great overthinking, but you need to think one more step… what if Felicity is just a common IT nerd? What if the standard IT job involves crafting rafts of nanobots and ever changing processing speed computers…
What does this say about the state of IT in the Arrow universe?

5. AndrewB #

First off, loved the article. I’m a sucker for any technological deconstruction in media. Thanks for writing it. That said…

“What could be making processors?”

Why make them? Felicity hacks into city, government, business, etc. computer networks every episode. Perhaps she’s setup a botnet across Starling City and Queen Consolidated. Her processing speed increases as she distributes that data across her network, much like [email protected] searches through radio signals. She’s not scaling up, she’s scaling out.