Flat Earth Mathematics with examples in the R Language

YouTube contains thousands of videos from people that claim the earth is flat. These people pontificate that science as we know it is a "globalist conspiracy". The claims about a flat earth are hilarious and entertaining. People who believe our planet is flat even conduct simple experiments and engage in naive flat earth mathematics. Needless to say that these experiments are not very convincing, but I respect their entrepreneurial attitude.

Adherents to the idea that the world is flat often propose Gleason's Map as their version of the correct representation of our planet. Every printed map of the earth is flat, but somehow this one seems to have a special status.

The Gleason map is not a map of a flat earth but a polar azimuthal equidistant projection of the globe. Even though the map itself states that it is a projection of the world, flat earth believers nevertheless see it as a literal representation of the world they live on.

Their belief is based on an appeal to common sense as the world appears to be flat when we look at the horizon. The second ingredient is a deep distrust in science, often inspired by religious motives.

This article shows two different ways to look at the earth and show that the spherical model better fits the reality of my trip around the world.

Projecting the spherical earth on a flat surface is a complex task. Maps require compromise as it is impossible to truthfully draw the surface of the globe on a piece of paper without distortion. The video below provides a practical introduction to map projections and shows how maps are stretched to display them on a flat surface.

The Azimuthal equidistant method is popularised in the flag of the United Nations. Antarctica is in this projection displayed as a ring around the world because the south pole projects to infinity. Flat earth evangelists believe that the South Pole a ring of ice that prevents us from proceeding beyond the disc. The edge of this disc cannot be shown because this method projects the south pole at an infinite distance from the centre of the map. We can recreate Gleason's map with ggplot, which incorporates the mapproj package to show maps in various projections.

geography

  ## Flat Earth Mathematics
  library(ggplot2)

  world <- map_data("world")

  ggplot(world) +
    geom_path(aes(x = long, y = lat, group = group), linewidth = .2) +
    theme(axis.text = element_blank(),
          axis.ticks = element_blank()) +
    labs(x = "", y = "") +
    coord_map("azequidistant", orientation = c(90, 0, 270))

The coord_map function allows you to change projections. The orientation argument defines the centre point of the projection. The most common location is the North Pole (90, 0). The third value gives the clockwise rotation in degrees. Gleason's map uses a rotation of 270 degrees.

In September 2018 I embarked on a trip around the world to speak at the World Water Congress in Tokyo. I also visited family and friends in the Netherlands and did some tourism in San Francisco. One of the workshops I speak at is about Fake News in the water industry.

My trip started in Australia, onward to Japan, from where I go to the Netherlands. The last two legs will bring me back to Australia via San Francisco. The itinerary is stored in a data frame. The ggmap package geocodes the longitude and latitude of each of the locations on my trip. You will need a Google API to enable the geocoding function. As the earth is a sphere, an intermediate point needs to be added for trips that pass the dateline, as I explained in my article about flight maps.

The geosphere package helps to estimate the total travel distance. The travel distance is approximately 38,995 km, slightly less than a trip around the equator. This distance is the great circle distance, which is the shortest distance between two points on a sphere. The flight paths on this map are curved because of the inevitable distortions when projecting a sphere on a flat surface.

  ## Round the World Itinerary
  library(ggmap)
  library(dplyr)

  api <- readLines("google.api") # Text file with the secret API key
  register_google(key = api)
  airports <- c("Melbourne", "Tokyo", "Amsterdam", "San Francisco")
  itinerary <- geocode(airports)
  itinerary <- rbind(itinerary, itinerary[1, ]) %>%
    mutate(location = c(airports, airports[1]))

  ## Split travel past dateline
  dl <- which(diff(itinerary$lon) > 180)
  dr <- ifelse(itinerary$lon[dl] < 0, -180, 180)
  dateline <- tibble(lon = c(dr, -dr),
                     lat = rep(mean(itinerary$lat[dl:(dl + 1)]), 2),
                     location = "dateline")
  itinerary <- rbind(itinerary[1:dl, ], dateline,
                     itinerary[(dl + 1):nrow(itinerary), ])
  itinerary

  ## Visualise
  worldmap +
    geom_point(data = itinerary, aes(lon, lat), colour = "red", size = 4) +
    geom_path(data = itinerary, aes(lon, lat), colour = "red", size = 1) +
    coord_map("azequidistant", orientation = c(90, 0, 270))

  ## Great Circle Distance
  library(geosphere)
  sapply(1:(nrow(itinerary) - 1), function(l)
    distVincentyEllipsoid(itinerary[l, 1:2], itinerary[(l + 1), 1:2]) / 1000) %>%
    sum()

If the Gleason map were an actual map of the flat earth, then the flight paths on the map would show as straight lines.

The mapproj package contains the mapproject function that calculates the projected coordinates based on longitude and latitude. The output of this function is a grid with limits from $-\pi$ to $\pi$. The first part of the code converts the longitude and latitude from the world data frame to projected coordinates.

A line from the lon/lat 0,0 to the north pole has a projected distance of \(\pi/2\), which in the spherical world is \(\frac{\pi}{2} 6378.137 = 10018.75\) km. We need to multiply the Euclidean distances with the radius of the Earth to derive the Gleason map coordinates.

This last code snippet converts the world map to flat earth coordinates. It calculates the Euclidean distance between the points on the itinerary and multiplies this with the Earth's diameter.

The best way to show that Gleason's map is not a map of a flat earth is best demonstrated with a flight between Australia and Chili because the distortions are huge in the Southern hemisphere. This last code snippet shows why the Gleason map is not a map of a flat earth. On this map, the shortest distance between Sydney and Santiago de Chili is about 25,000 km, more than twice the real value. The actual travel time is about 14 hours, which implies that passenger jets break the sound barrier. This problem exists for all journeys along the lines of latitude in the Southern hemisphere.

This map looks like the first one, but the coordinate system is now Euclidean instead of polar (longitudes and latitudes), as indicated by the square grid. On a projected map, the shortest distance is a curved line, parallel to Antarctica, which is how ships and aeroplanes move between these cities.

This video shows a unique insight into the navigational issues for intercontinental flights in the Southern hemisphere.

  library(mapproj)
  flatearth.coords <- mapproject(world$long, world$lat,
                          "azequidistant", orientation = c(90, 0, 270))
  r <- 6378.137
  flatearth.coords <- mutate(world,
                  x = flatearth.coords$x * r,
                  y = flatearth.coords$y * r) %>%
      select(x, y, group, order, region, subregion)

  flatearth <- ggplot(flatearth.coords) +
      geom_path(aes(x, y, group = group), size = .2) +
      theme(axis.text = element_blank(),
            axis.ticks = element_blank()) +
      labs(x = "", y = "")

  ## Australia - South America
  airports <- tibble(city = c("Sydney", "Santiago de Chili"))
  itinerary <- geocode(airports$city) %>%
      mutate(location = airports$city)
  itinerary
  coords <- mapproject(itinerary$lon, itinerary$lat, "azequidistant",
                       orientation = c(90, 0, 270))
  coords <- tibble(x = coords$x * r, y = coords$y * r)
  sum(sqrt(diff(coords$x)^2 + diff(coords$y)^2))

  flatearth + 
      geom_point(data = coords, aes(x, y), colour = "red", size = 4) +
      geom_path(data = coords, aes(x, y), colour = "red", size = 1)

geography

This article proves that the Gleason map is not a representation of a flat earth. Aeroplanes would have to break the sound barrier to fly these distances in the time it takes to travel. Whichever way you project the globe on a flat map will lead to inevitable distortions. The Gleason map itself mentions that it is a projection and not a flat earth. However, these facts will not dissuade people from believing in a flat earth. I am after all an engineer and thus part of the globalist science conspiracy.

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