The Meeting Point Locator

Hi Hillary, It’s Donald, would you like to have a beer with me in La Cabra Brewing, in Berwyn, Pensilvania? (Hypothetical utilization of The Meeting Point Locator)

Finding a place to have a drink with someone may become a difficult task. It is quite common that one of them does not want to move to the other’s territory. I am sure you have faced to this situation many times. With The Meeting Point Locator this will be no longer an issue, since it will give you a list of equidistant bars and coffees to any two given locations. Let’s see an example.

I do not know if Hillary Clinton and Donald Trump have met each other after the recent elections in United States, but the will probably do. Let’s suppose Hillary doesn’t want to go to The White House and that Donald prefers another place instead Hillary’s home. No problem at all. According to this, Hillary lives in Chappaqua, New York and Donald will live in The White House, Washington (although he supposedly won’t do full time as he announced recently). These two locations are the only input that The Meeting Point Locator needs to purpose equidistant places where having a drink. This is how it works:

  • Generates a number of coordinates on the great circle which passes through the midpoint of the original locations and is orthogonal to the rhumb defined by them; the number of points depends on the distance between the original locations.
  • Arranges these coordinates according to the distance to the original locations, from the nearest to the most distant.
  • Depending also on the distance of the original locations, defines a radius to search around each point generated on the great circle (once calculated, this radius is constant for all searches).
  • Starting from the nearest point, looks for a number of places (20 by default) to have a drink using the radius calculated previously. To do this, it calls to the Google Places API. Once the number of locations is reached, the proccess stops.

This map shows the places purposed for Hillary and Donald (blue points) as well as the original locations (red ones). You can make zoom in for details:

These are the 20 closest places to both of them:

listHillaryTrumpDT

Some other examples of the utility of The Meeting Point Locator:

  • Pau Gasol (who lives in San Antonio, Texas) and Marc Gasol (in Memphis, Tennessee) can meet in The Draft Sports Bar, in Leesville (Louisiana) to have a beer while watching a NBA match. It is 537 kilometers far from both of them.
  • Bob Dylan (who lives in Malibu, California) and The Swedish Academy (placed in Stockholm, Sweden) can smooth things over drinking a caipirinha in Bar São João, in Tremedal (Brasil)only 9.810 kilometers far from both of them.
  • Spiderman (placed in New York City) and Doraemon (in Tokio, Japan) can meet in Andreyevskaya, in Stroitel (Russia) to have a have a hot drink. Since they are superheroes, they will cover the 9.810 kilometers of separation in no time at all.

I faced with two challenges to do this experiment: how to generate the orthogonal great circle from two given locations and how to define radius and number of points over this circle to do searchings. I will try to explain in depth both things in the future in another post.

You will find the code below. To make it work, do not forget to get your own key for Google Places API Web Service here. I hope this tool will be helpful for someone; if yes, do not hesitate to tell it to me.

library(httr)
library(jsonlite)
library(dplyr)
library(ggmap)
library(geosphere)
library(DT)
library(leaflet)

# Write both addresses here (input)
place1="Chappaqua, New York, United States of America"
place2="The White House, Washington DC, United States of America"

# Call to Google Maps API to obtain coordinates of previous addresses
p1=geocode(place1, output = "latlon")
p2=geocode(place2, output = "latlon")

# To do searchings I need a radius
radius=ifelse(distGeo(p1, p2)>1000000, 10000,
              ifelse(distGeo(p1, p2)>100000, 2500, 1000))

# And a number of points
npoints=ifelse(distGeo(p1, p2)>1000000, 2002,
               ifelse(distGeo(p1, p2)>100000, 7991, 19744))

# Place here the Google Places API Key
key="PLACE_YOUR_OWN_KEY_HERE"

# Build the url to look for bars and cafes with the previous key
url1="https://maps.googleapis.com/maps/api/place/nearbysearch/json?location=lat,lon&radius="
url2="&types=cafe|bar&key="
url=paste0(url1,radius,url2,key)

# This is to obtain the great circle orthogonal to the rhumb defined by input locations
# and which passes over the midpoint. I will explain this step in the future
mid=midPoint(p1, p2)
dist=distGeo(p1, p2)
x=p1
y=p2
while(dist>1000000)
{
  x=midPoint(mid, x)
  y=midPoint(mid, y)
  dist=distGeo(x, y)
}

bea=bearingRhumb(x, y)
points=greatCircle(destPoint(p=mid, b=bea+90, d=1), mid, n=npoints)

# Arrange the points dependning on the distance to the input locations
data.frame(dist2p1=apply(points, 1, function (x) distGeo(p1, x)),
           dist2p2=apply(points, 1, function (x) distGeo(p2, x))) %>% 
  mutate(order=apply(., 1, function(x) {max(x)})) %>% 
  cbind(points) %>% 
  arrange(order) -> points

# Start searchings
nlocs=0 # locations counter (by default stops when 20 is reached)
niter=1 # iterations counter (if greater than number of points on the great circle, stops)
results=data.frame()
while(!(nlocs>=20 | niter>npoints)) {
  print(niter)
  url %>% 
    gsub("lat", points[niter, 'lat'], .) %>% 
    gsub("lon", points[niter, 'lon'], .) %>% 
    GET %>% 
    content("text") %>% 
    fromJSON -> retrieve
  
  df=data.frame(lat=retrieve$results$geometry$location$lat,
                lng=retrieve$results$geometry$location$lng,
                name=retrieve$results$name, 
                address=retrieve$results$vicinity)
  results %>% rbind(df)->results
  
  nlocs=nlocs+nrow(df)
  niter=niter+1 
}

# I prepare results to do a Data Table
data.frame(dist2p1=apply(results, 1, function (x) round(distGeo(p1, c(as.numeric(x[2]), as.numeric(x[1])))/1000, digits=1)),
           dist2p2=apply(results, 1, function (x) round(distGeo(p2, c(as.numeric(x[2]), as.numeric(x[1])))/1000, digits=1))) %>% 
  mutate(mx=apply(., 1, function(x) {max(x)})) %>% 
  cbind(results) %>% 
  arrange(mx) %>% 
  mutate(rank=row_number()) %>% 
  select(-mx)-> resultsDT

# This is the Data table
datatable(resultsDT, 
          class = 'cell-border stripe',
          rownames = FALSE,
          options = list(pageLength = 5),
          colnames = c('Distance to A (Km)', 
                       'Distance to B (Km)', 
                       'Latitude', 
                       'Longitude',
                       'Name', 
                       'Address', 
                       'Rank'))

# Map with the locations using leaflet
resultsDT %>% 
  leaflet() %>% 
  addTiles() %>% 
  addCircleMarkers(
    lng=resultsDT$lng, lat=resultsDT$lat,
    radius = 8,
    color = "blue",
    stroke = FALSE, fillOpacity = 0.5,
    popup=paste(paste0("<b>", resultsDT$name, "</b>"), resultsDT$address, sep="
")
  ) %>% 
  addCircleMarkers(
    lng=p1$lon, lat=p1$lat,
    radius = 10,
    color = "red",
    stroke = FALSE, fillOpacity = 0.5,
    popup=paste("<b>Place 1</b>", place1, sep="
")
  )%>% 
  addCircleMarkers(
    lng=p2$lon, lat=p2$lat,
    radius = 10,
    color = "red",
    stroke = FALSE, fillOpacity = 0.5,
    popup=paste("<b>Place 2</b>", place2, sep="
")
  )

Three Shiny Apps to Celebrate the Beauty of Maths

Mathematics knows no races or geographic boundaries; for mathematics, the cultural world is one country (David Hilbert)

One of the best decisions I took this year related with this blog was to move it to my own self-hosted domain using WordPress.org. It allows to me, for example, to embed dynamic JavaScript visualizations like this one. Another thing I can do now is to upload my Shiny Apps to share them with my readers. In this post I have gathered three Apps I made some time ago; you can play with them as well as get the code I wrote for each one:

  • The Harmonograph: This App simulates harmonograph drawings. An harmonograph is a mechanism which draws trajectories by means of two pendulums: one moves a pencil and the other one moves a platform with a piece of paper on it. Click here to try it.
  • Shiny Wool Skeins: This App, inspired by this post, creates a plot consisting of chords inside a circle . You can change colors as well as the number and quality of the chords. Click here to try it.
  • The Coaster Maker: With this App you can create your own coasters using hypocicloids. Click here to try it.

I want to thank to my friend Jorge, without whom I would not have been able to make Shiny work in my server.

The Somnambulist and Pi

How wary we are of something warm and unborn. Something calmly by zero will divide (Unbegotten, The Somnambulist)

Some time ago, I assumed the mission to draw a plot for the cover of the new album of The Somnambulist, a music band from Berlin. They wanted a circlization of Pi, which is a graphic where numbers are represented in a circular layout. The idea is connecting each digit of Pi to its successive digit with links to the position of the numerically corresponding external sectors. I used a color palette composed by 10 nuances of the visible spectrum as a tribute for Planck, as Marco (the vocalist) requested me. After a number of attempts:

somnambulist

The album is named Unbegotten, a german word which means archaic. As Marco told me, in theology it also means kind of eternal because of being never born and so never dying. I like how π is integrated into the title to substitute the string “tt” in the middle. Pi is also eternal so the association is genuine.

The music of The Somnambulist is intense, dark and powerful and is waiting for you here to listen it. My favorite song is the one that gives name to the album.

If you want to know more about circlizong numbers, you can visit this post, where you also can see the code I used as starting point to do this plot.

Bayesian Blood

The fourth, the fifth, the minor fall and the major lift (Hallelujah, Leonard Cohen)

Next problem is extracted from MacKay’s Information Theory, Inference and Learning Algorithms:

Two people have left traces of their own blood at the scene of a crime. A suspect, Oliver, is tested and found to have type ‘O’ blood. The blood groups of the two traces are found to be of type ‘O’ (a common type in the local population, having frequency 60%) and of type ‘AB’ (a rare type, with frequency 1%). Do these data give evidence in favor of the proposition that Oliver was one of the people who left blood at the scene?

To answer the question, let’s first remember the probability form of Bayes theorem:

p(H|D)=\dfrac{p(H)p(D|H)}{p(D)}

where:

  • p(H) is the probability of the hypothesis H before we see the data, called the prior
  • p(H|D) is the probablity of the hyothesis after we see the data, called the posterior
  • p(D|H) is the probability of the data under the hypothesis, called the likelihood
  • p(D)is the probability of the data under any hypothesis, called the normalizing constant

If we have two hypothesis, A and B, we can write the ratio of posterior probabilities like this:

\dfrac{p(A|D)}{p(B|D)}=\dfrac{p(A)p(D|A)}{p(B)p(D|B)}

If p(A)=1-p(B) (what means that A and B are mutually exclusive and collective exhaustive), then we can rewrite the ratio of the priors and the ratio of the posteriors as odds. Writing o(A) for odds in favor of A, we get the odds form of Bayes theorem:

o(A|D)=o(A)\dfrac{p(D|A)}{p(D|B)}

Dividing through by o(A) we have:

\dfrac{o(A|D)}{o(A)}=\dfrac{p(D|A)}{p(D|B)}

The term on the left is the ratio of the posteriors and prior odds. The term on the right is the likelihood ratio, also called the Bayes factor. If it is greater than 1, that means that the data were more likely under A than under B. And since the odds ratio is also greater than 1, that means that the odds are greater, in light of the data, than they were before. If the Bayes factor is less than 1, that means the data were less likely under A than under B, so th odds in favor of A go down.

Let’s go back to our initial problem. If Oliver left his blood at the crime scene, the probability of the data is just the probability that a random member of the population has type ‘AB’ blood, which is 1%. If Oliver did not leave blood at the scene, what is the the chance of finding two people, one with type ‘O’ and one with type ‘AB’? There are two ways it might happen: the first person we choose might have type ‘O’ and the second ‘AB’, or the other way around. So the probability in this case is 2(0.6)(0.01)=1.2%. Dividing probabilities of both scenarios we obtain a Bayes factor of 0.83, and we conclude that the blood data is evidence against Oliver’s guilt.

Once I read this example, I decided to replicate it using real data of blood type distribution by country from here. After cleaning data, I have this nice data set to work with:

For each country, I get the most common blood type (the one which the suspect has) and the least common and replicate the previous calculations. For example, in the case of Spain, the most common type is ‘O+’ with 36% and the least one is ‘AB-‘ with 0.5%. The Bayes factor is 0.005/(2(0.36)(0.005))=1.39 so data support the hypothesis of guilt in this case. Next chart shows Bayes factor accross countries:

Just some comments:

  • Sometimes data consistent with a hypothesis are not necessarily in favor of the hypothesis
  • How different is the distribution of blood types between countries!
  • If you are a estonian ‘A+’ murderer, choose carefully your accomplice

This is the code of the experiment:

library(rvest)
library(dplyr)
library(stringr)
library(DT)
library(highcharter)

# Webscapring of the table with the distribution of blood types
url <- "http://www.rhesusnegative.net/themission/bloodtypefrequencies/"
blood <- url %>%
   read_html() %>%
   html_node(xpath='/html/body/center/table') %>%
   html_table(fill=TRUE)

# Some data cleansing
blood %>% slice(-c(66:68)) -> blood

blood[,-c(1:2)] %>% 
  sapply(gsub, pattern=",", replacement=".") %>% 
  as.data.frame %>% 
  sapply(gsub, pattern=".79.2", replacement=".79") %>% 
  as.data.frame-> blood[,-c(1:2)]

blood %>% 
  sapply(gsub, pattern="%|,", replacement="") %>% 
  as.data.frame -> blood

blood[,-1] = apply(blood[,-1], 2, function(x) as.numeric(as.character(x)))


blood[,-c(1:2)] %>% mutate_all(funs( . / 100)) -> blood[,-c(1:2)]

# And finally, we have a nice data set
datatable(blood, 
          rownames = FALSE,
          options = list(
          searching = FALSE,
          pageLength = 10)) %>% 
  formatPercentage(3:10, 2)

# Calculate the Bayes factor
blood %>% 
  mutate(factor=apply(blood[,-c(1,2)], 1, function(x) {min(x)/(2*min(x)*max(x))})) %>% 
  arrange(factor)-> blood

# Data Visualization
highchart() %>% 
     hc_chart(type = "column") %>% 
     hc_title(text = "Bayesian Blood") %>%
     hc_subtitle(text = "An experiment about the Bayes Factor") %>%  
     hc_xAxis(categories = blood$Country, 
             labels = list(rotation=-90, style = list(fontSize = "12px")))  %>% 
     hc_yAxis(plotBands = list(list(from = 0, to = 1, color = "rgba(255,215,0, 0.8)"))) %>% 
     hc_add_series(data = blood$factor,
                   color = "rgba(255, 0, 0, 0.5)",
                   name = "Bayes Factor")%>% 
  hc_yAxis(min=0.5) %>% 
  hc_tooltip(pointFormat = "{point.y:.2f}") %>% 
  hc_legend(enabled = FALSE) %>% 
  hc_exporting(enabled = TRUE) %>%
  hc_chart(zoomType = "xy")

Visualizing the Daily Variability of Bitcoin with Quandl and Highcharts

Lay your dreams, little darling, in a flower bed; let that sunshine in your hair (Where the skies are blue, The Lumineers)

I discovered this nice visualization some days ago. The author is also the creator of Highcharter, an incredible R wrapper for Highcharts javascript libray and its modules. I am a big fan of him.

Inspired by his radial plot, I did a visualization of the daily evolution of Daily Bitcoin exchange rate (BTC vs. EUR) on Localbtc. Data is sourced from here and I used Quandl to obtain the data frame. Quandl is a marketplace for financial and economic data delivered in modern formats for today’s analysts. There is a package called Quandl to interact directly with the Quandl API to download data in a number of formats usable in R. You only need to locate the data you want in the Quandl site. In my case data are here.

After loading data, I do the folowing steps:

  • Filtering data to obtain last 12 complete months
  • Create a new variable with the difference between closing and opening price of Bitcoin (in Euros)
  • Create a color variable to distinguish between positive and negative differences
  • Create the graph using Fivethirtyeight theme for highcharts

This is the result:

Apart of its appealing, I think is a good way to to have a quick overview of the evolution of a stock price. This is the code to do the experiment:

library(Quandl)
library(dplyr)
library(highcharter)
library(lubridate)
bitcoin=Quandl("BCHARTS/LOCALBTCEUR")
bitcoin %>% 
  arrange(Date) %>% 
  mutate(tmstmp = datetime_to_timestamp(Date)) -> bitcoin
last_date=max(bitcoin$Date)
if (day(last_date+1)==1) date_to=last_date else 
  date_to=ymd(paste(year(last_date), month(last_date),1, sep="-"))-1
date_from=ymd(paste(year(date_to)-1, month(date_to)+1,1, sep="-"))
bitcoin %>% filter(Date>=date_from, Date<=date_to) -> bitcoin
var_bitcoin <- bitcoin %>% 
  mutate(Variation = Close - Open,
         color = ifelse(Variation>=0, "green", "red"),
         y = Variation) %>% 
  select(x = tmstmp,
         y,
         variation = Variation,
         name = Date,
         color,
         open = Open,
         close = Close) %>% 
  list.parse3()
x <- c("Open", "Close", "Variation")
y <- sprintf("{point.%s}", tolower(x))
tltip <- tooltip_table(x, y)
hc <- highchart() %>% 
  hc_title(text = "Bitcoin Exchange Rate (BTC vs. EUR)") %>% 
  hc_subtitle(text = "Daily Variation on Localbtc. Last 12 months")%>% 
  hc_chart(
    type = "column",
    polar = TRUE) %>%
  hc_plotOptions(
    series = list(
      stacking = "normal",
      showInLegend = FALSE)) %>% 
  hc_xAxis(
    gridLineWidth = 0.5,
    type = "datetime",
    tickInterval = 30 * 24 * 3600 * 1000,
    labels = list(format = "{value: %b}")) %>% 
  hc_yAxis(showFirstLabel = FALSE) %>% 
  hc_add_series(data = var_bitcoin) %>% 
  hc_add_theme(hc_theme_538()) %>% 
  hc_tooltip(useHTML = TRUE,
    headerFormat = as.character(tags$small("{point.x:%d %B, %Y}")),
    pointFormat = tltip)
hc

Chaotic Galaxies

Tell me, which side of the earth does this nose come from? Ha! (ALF)

Reading about strange attractors I came across with this book, where I discovered a way to generate two dimensional chaotic maps. The generic equation is pretty simple:

x_{n+1}= a_{1}+a_{2}x_{n}+a_{3}x_{n}^{2}+a_{4}x_{n}y_{n}+a_{5}y_{n}+a_{6}y_{n}^{2}
y_{n+1}= a_{7}+a_{8}x_{n}+a_{9}x_{n}^{2}+a_{10}x_{n}y_{n}+a_{11}y_{n}+a_{12}y_{n}^{2}

I used it to generate these chaotic galaxies:

Changing the vector of parameters you can obtain other galaxies. Do you want to try?

library(ggplot2)
library(dplyr)
#Generic function
attractor = function(x, y, z)
{
  c(z[1]+z[2]*x+z[3]*x^2+ z[4]*x*y+ z[5]*y+ z[6]*y^2, 
    z[7]+z[8]*x+z[9]*x^2+z[10]*x*y+z[11]*y+z[12]*y^2)
}
#Function to iterate the generic function over the initial point c(0,0)
galaxy= function(iter, z)
{
  df=data.frame(x=0,y=0)
  for (i in 2:iter) df[i,]=attractor(df[i-1, 1], df[i-1, 2], z)
  df %>% rbind(data.frame(x=runif(iter/10, min(df$x), max(df$x)), 
                          y=runif(iter/10, min(df$y), max(df$y))))-> df
  return(df)
}
opt=theme(legend.position="none",
          panel.background = element_rect(fill="#00000c"),
          plot.background = element_rect(fill="#00000c"),
          panel.grid=element_blank(),
          axis.ticks=element_blank(),
          axis.title=element_blank(),
          axis.text=element_blank(),
          plot.margin=unit(c(-0.1,-0.1,-0.1,-0.1), "cm"))
#First galaxy
z1=c(1.0, -0.1, -0.2,  1.0,  0.3,  0.6,  0.0,  0.2, -0.6, -0.4, -0.6,  0.6)
galaxy1=galaxy(iter=2400, z=z1) %>% ggplot(aes(x,y))+
  geom_point(shape= 8, size=jitter(12, factor=4), color="#ffff99", alpha=jitter(.05, factor=2))+
  geom_point(shape=16, size= jitter(4, factor=2), color="#ffff99", alpha=jitter(.05, factor=2))+
  geom_point(shape=46, size= 0, color="#ffff00")+opt
#Second galaxy
z2=c(-1.1, -1.0,  0.4, -1.2, -0.7,  0.0, -0.7,  0.9,  0.3,  1.1, -0.2,  0.4)
galaxy2=galaxy(iter=2400, z=z2) %>% ggplot(aes(x,y))+
  geom_point(shape= 8, size=jitter(12, factor=4), color="#ffff99", alpha=jitter(.05, factor=2))+
  geom_point(shape=16, size= jitter(4, factor=2), color="#ffff99", alpha=jitter(.05, factor=2))+
  geom_point(shape=46, size= 0, color="#ffff00")+opt
#Third galaxy
z3=c(-0.3,  0.7,  0.7,  0.6,  0.0, -1.1,  0.2, -0.6, -0.1, -0.1,  0.4, -0.7)
galaxy3=galaxy(iter=2400, z=z3) %>% ggplot(aes(x,y))+
  geom_point(shape= 8, size=jitter(12, factor=4), color="#ffff99", alpha=jitter(.05, factor=2))+
  geom_point(shape=16, size= jitter(4, factor=2), color="#ffff99", alpha=jitter(.05, factor=2))+
  geom_point(shape=46, size= 0, color="#ffff00")+opt
#Fourth galaxy
z4=c(-1.2, -0.6, -0.5,  0.1, -0.7,  0.2, -0.9,  0.9,  0.1, -0.3, -0.9,  0.3)
galaxy4=galaxy(iter=2400, z=z4) %>% ggplot(aes(x,y))+
  geom_point(shape= 8, size=jitter(12, factor=4), color="#ffff99", alpha=jitter(.05, factor=2))+
  geom_point(shape=16, size= jitter(4, factor=2), color="#ffff99", alpha=jitter(.05, factor=2))+
  geom_point(shape=46, size= 0, color="#ffff00")+opt

Gummy Worms

Just keep swimming (Dory in Finding Nemo)

Inspired by this post, I decided to create gummy worms like this:

worm
Or these:

When I was young I used to eat them.

Do you want to try? This is the code:

library(rgl)
library(RColorBrewer)
t=seq(1, 6, by=.04)
f = function(a, b, c, d, e, f, t) exp(-a*t)*sin(t*b+c)+exp(-d*t)*sin(t*e+f)
v1=runif(6,0,1e-02)
v2=runif(6, 2, 3)
v3=runif(6,-pi/2,pi/2)
open3d()
spheres3d(x=f(v1[1], v2[1], v3[1], v1[4], v2[4], v3[4], t),
          y=f(v1[2], v2[2], v3[2], v1[5], v2[5], v3[5], t),
          z=f(v1[3], v2[3], v3[3], v1[6], v2[6], v3[6], t),
          radius=.3, color=sample(brewer.pal(8, "Dark2"),1))

Visualizing the Gender of US Senators With R and Highmaps

I wake up every morning in a house that was built by slaves (Michelle Obama)

Some days ago I was invited by the people of Highcharts to write a post in their blog. What I have done is a simple but revealing map of women senators of the United States of America. Briefly, this is what I’ve done to generate it:

  • read from the US senate website a XML file with senators info
  • clean and obtain gender of senators from their first names
  • summarize results by state
  • join data with a US geojson dataset to create the highmap

You can find details and R code here.

It is easy creating a highcharts using highcharter, an amazing library as genderizeR, the one I use to obtain gender names. I like them a lot.

Visualizing Stirling’s Approximation With Highcharts

I said, “Wait a minute, Chester, you know I’m a peaceful man”, He said, “That’s okay, boy, won’t you feed him when you can” (The Weight, The Band)

It is quite easy to calculate the probability of obtaining the same number of heads and tails when tossing a coin N times, and N is even. There are 2^{N} possible outcomes and only C_{N/2}^{N} are favorable so the exact probability is the quotient of these numbers (# of favorable divided by # of possible).

There is another way to approximate this number incredibly well: to use the Stirling’s formula, which is 1/\sqrt{\pi\cdot N/2}

The next plot represents both calculations for N from 2 to 200. Although for small values of N, Stirling’s approximation tends to overestimate probability, you can see hoy is extremely precise as N becomes bigger:

James Stirling published this amazing formula in 1730. It simplifies the calculus to the extreme and also gives a quick way to obtain the answer to a very interesting question: how many tosses are needed to be sure that the probability of obtaining the same number of heads and tails is under any given threshold? Just solve the formula for N and you will obtain the answer. And, also, the formula is another example of the presence of pi in the most unexpected places, as happens here.

Just another thing: the more I use highcharter package the more I like it.

This is the code:

library(highcharter)
library(dplyr)
data.frame(N=seq(from=2, by=2, length.out = 100)) %>%
  mutate(Exact=choose(N,N/2)/2**N, Stirling=1/sqrt(pi*N/2))->data
hc <- highchart() %>% 
  hc_title(text = "Stirling's Approximation") %>% 
  hc_subtitle(text = "How likely is getting 50% heads and 50% tails tossing a coin N times?") %>% 
  hc_xAxis(title = list(text = "N: Number of tosses"), categories = data$N) %>% 
  hc_yAxis(title = list(text = "Probability"), labels = list(format = "{value}%", useHTML = TRUE)) %>% 
  hc_add_series(name = "Stirling", data = data$Stirling*100,  marker = list(enabled = FALSE), color="blue") %>% 
  hc_add_series(name = "Exact", data = data$Exact*100,  marker = list(enabled = FALSE), color="lightblue") %>% 
  hc_tooltip(formatter = JS("function(){return ('<b>Number of tosses: </b>'+this.x+'
<b>Probability: </b>'+Highcharts.numberFormat(this.y, 2)+'%')}")) %>%
  hc_exporting(enabled = TRUE) %>%
  hc_chart(zoomType = "xy")
hc