A Gaussian KDE is a sum of little Gaussians

Today I learned from here that the Gaussian KDE is the sum of a bunch of little Gaussian curves. Here look:

library(tidyverse)
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library(palmerpenguins)
#> 
#> Attaching package: 'palmerpenguins'
#> 
#> The following objects are masked from 'package:datasets':
#> 
#>     penguins, penguins_raw

set.seed(20211122)
d <- penguins %>% 
  filter(!is.na(bill_length_mm)) %>% 
  sample_n(size = 20)

# grid of xs
x <- seq(30, 63, length.out = 200)

# compute density of xs using each observed value
# as the mean.
l <- lapply(
  d$bill_length_mm,
  function(m) {
    data.frame(
      x = x,
      y = dnorm(x, mean = m, sd = bw.nrd0(d$bill_length_mm)) 
    )
  }
) 
dl <- bind_rows(l, .id = "obs")

# plot them, their sum, and the default density curve
ggplot(d) + 
  aes(x = bill_length_mm) + 
  geom_density(aes(color = "geom_density()"), size = 2, key_glyph = "path") +
  geom_rug() +
  geom_line(
    aes(x = x, y = y / nrow(d), group = obs),
    data = dl, alpha = .1
  ) + 
  stat_summary(
    aes(x = x, y = y / nrow(d), color = "sum of pointwise densities"),
    data = dl, 
    # alpha = .1, 
    geom = "line",
    fun = "sum", 
    # color = "orange"
  ) +
  scale_color_manual(values = c("black", "orange")) +
  labs(color = NULL)
#> Warning: Using `size` aesthetic for lines was deprecated in ggplot2 3.4.0.
#> ℹ Please use `linewidth` instead.
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#> generated.

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