Verum Tinfoil Oneshot: How thick is Verum’s booty?
neoshadow
02/10/2021
How thick must literal Booty of the Verum be?
WARNING: This also not canon and just for fun and being lewd with math.
I really love Verum by the Numbers because it was fun and silly way to speculate about the Verum universe, using science but without weight of saying “canon”. With the author Gen Nova on a break, I wanted to do something in honor of all the great math art that Gen Nova did. With the Crowns going on a trip through the astral sea, anyone might be able to chart there possible path across the astral sea. Specifically how high and long they traveled. This begs an interesting question… How thick are the Verum petal nebulas? Verum by the numbers used an assumption of how long the glorious space flower is and we know how wide it is, but do not know how ROUND the supple curves are. As Scatter Clowns penetrates the Verum and Abstraction of General Relativity to pure wormhole math, we can measure the minimum path they take and to estimate the minimal depth of Verum’s cheeks.
Edit 02/10/2020
This math is dedicated to the SummerSalt. I probably would have not finished this, without you and LW groups constant wholesome DnD. Euclidean Math Abstraction is hard. Hugs!!
Its time to lewd all numbers…
We know how wide and long the Verum Galaxy is, but how thick is her celestial booty?
Setup of the Equations used:
For some of the math (Case 3 and Case 4), I will be using a more general form distance than Pythagorean Triangles allows called Minkowski Distance (aka Minkowski Displacement), which has some pretty cool properties from a DnD perspective especially if players dislike Pythagorean Triangles.
Its probably good to at least mention the equation, since it very easy to get confused with related but less general equation called Minkowksi Spacetime. For some infinite number of dimensions \(n\) (containing \(x,y,z,t, ... etc.\) and dummy indexes \(X,Y\)) the Minkowski distance \(d\) between two points from \(s\) (for start) to \(f\) (for finish) is… \[ d(s_X,f_Y,p)=\left | \sum_{i=1}^{n}\left (f_{i}-s_{i} \right )^p \right |^{\tfrac{1}{p}} \] where the following is defined:
- The n dimensions \(n=(x,y,z,t,... n) \in N\). This is just a math formalism, but n-dimension is dimensional vector \(n\) and includes x,y,z,t. The dimensions are counted using integers. Also \(s_X\) and \(f_Y\) are subsets of \(n\).
- The start point is \(s_X=(s_x,s_y,s_z,s_t, ... s_n)=(x_s,y_s,z_s,t_s, ... n_s) \in \mathbb{C}^n\)
- The finish point is \(f_Y=(f_x,f_y,f_z,f_t, ... f_n)=(x_f,y_f,z_f,t_f, ... n_f) \in \mathbb{C}^n\)
- The parameter \(p\) is the space parameter is a parameter of what kind of curvature space has and how curved that space can be. For Minkowski distance, \(1 \leq p \leq 2\). It is also know as p-norm which comes from the math of \(L^p\) spaces.
Euclidean distance (2D,3D) when p=2
When \(p=2\), we have Euclidean Geometry and regular spacetime (Minowski Spacetime). \[ d(s_X,f_Y,p=2)=\left | \sum_{i=1}^{n}\left (f_{i}-s_{i} \right )^2 \right |^{\tfrac{1}{2}}=\sqrt{\sum_{i=1}^{n}\left( f_{i}-s_{i} \right )^2} \] \[d(s_2,f_2,p=2)=\sqrt{\sum_{i=1}^{2}\left( f_{i}-s_{i} \right )}=\sqrt{(x_f-x_s)^2+(y_f-y_s)^2}\] \[ d(s_3,f_3,p=2)=\sqrt{\sum_{i=1}^{2}\left( f_{i}-s_{i} \right )}=\sqrt{(x_f-x_s)^2+(y_f-y_s)^2+(z_f-z_s)^2} \]
Minkowski Spacetime metric (4D from General and Special Relativity):
\[ \begin{Bmatrix} Constant=x^2+y^2+z^2+(ict)^2 \\ \delta\tau = \delta x^2+\delta y^2+\delta x^2+(ic\delta t)^2 \end{Bmatrix} \] \[ d(s4,s4,p=2)=\sqrt{\sum_{i=1}^{4}\left( f_{i}-s_{i} \right )} =\sqrt{\delta x^2+\delta y^2+\delta x^2+(ic\delta t)^2}=\sqrt{(x_f-x_s)^2+(y_f-y_s)^2+(z_f-z_s)^2-c^2(t_f-t_s)^2} \] For you fellow science nerds out there, this math could allow me to also generalize spacetime Tensors (like Riemann curvature tensor, Gravitational tensor, Super Minkowski space, super-symmetry). I am a firm believer that good fiction should not follow hard physics or astrophysics.
Mahattan Distance when p=1
When \(p=1\), we have something know as Mahattan Distance where the shortest distance between two points is the absolute value difference between two points. In other words the only valid distance moves are like those in chess–up, down, left, right, or diagonal (if you rotate the board). If you think about it Manhattan Distance is exactly how distance works while in DnD combat, except you rotate the board to go 45 degrees 
Here are equations that govern it.
\[ d(s_X,f_Y,p=1)=\left | \sum_{i=1}^{n}\left (f_{i}-s_{i} \right )^1 \right |^{\tfrac{1}{1}}=\sum_{i=1}^{n}{\left | f_i-s_i\right |} \] In 2d: \[ d(s_2,f_2,p=1)=\left |\sum_{i=1}^{n}{ f_i-s_i}\right |=|(x_f-x_s)+(y_f-y_s)| \] In 3d: \[ d(s_2,f_2,p=1)=\left |\sum_{i=1}^{n}{ f_i-s_i}\right |=|(x_f-x_s)+(y_f-y_s)+(z_f-z_s)| \] In 4D (so called Mahattan Space Time): \[ d(s_4,f_4,p=1)=\left |\sum_{i=1}^{n}{ f_i-s_i}\right |\approx |(x_f-x_s)+(y_f-y_s)+(z_f-z_s)-c(t_f-t_s)| \]
Setup:
The overall trip can be summarized by the following simplified picture that shows both the Crowns journey and the Astral Traveller’s Path picture:
CurrentJourneyMap
The total fuel distance is 425 units. From Scattered Crowns, here is the fuel breakdown by Jump. The distance pixel
#for pretty tables
library(kableExtra)
#for easy and transparent code. %>% means "and then". SO its like SQL code.
library(tidyverse) ## -- Attaching packages --------------------------------------- tidyverse 1.3.0 --## v ggplot2 3.3.2 v purrr 0.3.4
## v tibble 3.0.4 v dplyr 1.0.2
## v tidyr 1.1.2 v stringr 1.4.0
## v readr 1.4.0 v forcats 0.5.0## -- Conflicts ------------------------------------------ tidyverse_conflicts() --
## x dplyr::filter() masks stats::filter()
## x dplyr::group_rows() masks kableExtra::group_rows()
## x dplyr::lag() masks stats::lag()library(readxl)
# library(astroR)
Fuel <- readxl::read_excel(path = "PlanetsTable.xlsx",sheet = "fuel")
kableExtra::kable(Fuel)| Jump (Roll) | Fuel Used | Location |
|---|---|---|
| 1 | 50 | Indigo |
| 2 | 130 | Indigo to Green |
| 3 | 130 | Green to Blue |
| 4 | 115 | Blue |
#Total Fuel is the sum of fuel
totalfuel = sum(Fuel$`Fuel Used`)
paste("The fuel is",totalfuel,"units.")## [1] "The fuel is 425 units."Case 2: Euclidean Distance
So I want to say in advance that I HATE Eucledian spherical trig. Technically if I publish this in an academic journel, I should also do the propogation of error (its the last one in the table), but I am not that crazy.
Lets also look Journey Map that is producted by tracing over Arcadum that they were supposed to follow.. sigh have a table of numbers….
EucleadanData <- readxl::read_excel(path = "PlanetsTable.xlsx",sheet = "segments")
#' Distance is found using Eucledian's formula + error because its not really 100% in 2D.
#' The error is in assuming that there is some perspective where the 3D space looks
#'almost 2D plus or minus an error term.
distance = sqrt((EucleadanData$x^2+EucleadanData$y^2+EucleadanData$`Error (kpc)`^2))
print(distance)## [1] 223.67032 261.70673 83.98458 139.64745 157.78913 16.41371 50.44215
## [8] 124.64915 47.88956 49.33974 301.07376Edistanceinparsec <- distance/670.4
Etotal = sum(Edistanceinparsec)
paste(Etotal, "kpc")## [1] "2.1727420526856 kpc"\[ d(s_2,f_2,p=1)=\left |\sum_{i=1}^{n}{ f_i-s_i}\right |=|(x_f-x_s)+(y_f-y_s)| \] Using the formula above, the total curvature of the trip is about 2.17 kpc which is slightly longer than the actually universe. This is an immediately hint that it has to be an extra 3D dimension.
MeanKParsecPerFuel=Etotal/totalfuel
ParsecPerFuel = MeanKParsecPerFuel*1000
print(ParsecPerFuel)## [1] 5.112334Every fuel it literally can jump 16.66. Now as noted in fuel table, it is more likely that the 3rd segment was the first two jumpts which was a total of 130+50=170 fuel. Estimating the flat or straight line distance is about 511.3 pixels but the path traveled 569.27
Using pathgorean theorm this means….
heightduetoEuler = sqrt(569.27*ParsecPerFuel^2-511.3*ParsecPerFuel^2)
paste(heightduetoEuler," parsecs")## [1] "38.9243070874789 parsecs"But wait there is more! This uses alot of assumptions about space and FTL travel. In fact, a more recent paper last week revealed that FTL might be more possible due to quantum mechanics. This paper probably would not happen, because required a quantum computer theorist and famous astrophysics to finally talk. It challenges some really basic high school physics assumptions. https://www.wired.co.uk/article/quantum-theory-speed-light-dragan
Case 3: Mahattan Distance
Now mahattan distance introduces an interesting concept. Because there are infinity paths, but have are actually all the same length if you think about it. Why is this possible? Wormhole FTL travel allow you to turn a corner in a straight linke
(Work in progress I could us an artist to show this visually
Let take the most visually intersting path the one were we have to meet at right angles, which means each distance can only be 45 jump is basically. The easiest path to think about is the one that is horizontal and then vertical at different angles.
Manhattan Distance Journey Picture
Case 4: Generalized Minkowski distance
The general equation is… \[ Eucldian Distance \leq Actual Distance \leq Mahattan Distance \]
We generalize the General Relativity to the maximum possible, then the actual thickness and all distances are truelly relative between two numbers. ;)