The 1974 Arecibo Radio Telescope Message

In 1974, the astronomer Frank Drake sent a math­e­mat­i­cal­ly-based bina­ry mes­sage to the stars (specif­i­cal­ly the M13 glob­u­lar clus­ter), using a radio tele­scope. What was the mes­sage, and how was it con­struct­ed?

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(L) The Areci­bo Mes­sage string, arranged in 73 rows and 23 columns. Even in text, your eye can see pat­terns emerg­ing. © The same mes­sage shown as a shad­ed grid, mak­ing the pat­terns more clear. ® The same image col­orized for dis­cus­sion. [Images by S. Lar­son; R image from Wiki­me­dia Com­mons]