From Vihart.

I saw a shape in a poster, and the rest could not be helped.

Want to make your own BTS Octahedron? http://vihart.com/bts/

Also see vihart.com/bts for more info, links, and original files that you can edit, translate, and reuse however you want.

This work is sponsored by Patreon Patrons like you! Special thanks to Ray Sidney, Pat Devlin, David Perryman, Andew Romaner, Andrea Bi Biagio, Jodi Vezzetti, Yana Chernobilsky, David Smith, and Caleb Wright.

For more paper octahedron fun, this intermediate origami octahedron by Jeremy Schafer is just wonderful: https://www.youtube.com/watch?v=rc5EGPo4Op0

One of the playing card constructions is "6 Card Ball" by Andrea Hawksley, she has DIY instructions here: http://blog.andreahawksley.com/six-card-ball/ The 12 card construction is by George Hart and has instructions here: http://georgehart.com/g4g10/ 

Also seen are a set of Skew Dice by Henry Segerman, as featured in this Numberphile video: https://www.youtube.com/watch?v=uAnCL3vhVIs

You may see a peek of a set of 4 truncated octahedra with one for each member of the Beatles. These came from the brilliant mind of an artist on Etsy named Tess, though she no longer seems to have a shop or a public-facing web presence. Tess, if you see this please please tell me WHAT WAS GOING THROUGH YOUR MIND WHEN YOU DECIDED BEATLES TRUNCATED OCTAHEDRA WERE A THING THAT NEEDED TO EXIST. Because you were right. So right. You give me faith that BTS Octahedra need to exist too.