Here is the full list of all 24 Abridge achievements.
Learn about Triangles
Complete all the Triangle Levels
Learn about Squares
Complete all the Square Levels
Learn about Diagonals
Complete all the Diagonal Levels
Learn about Circles
Complete all the Circle Levels
Complete all the Trio Levels
Complete all the Quad Levels
Complete half of the Trio Levels
Complete half of the Quad Levels
Learn about Mirrored Pieces
Complete all the Mirrored Levels
Learn about Corrupt Pieces
Complete all the Corrupt Levels
Learn about X's
Complete all the X Levels
Complete half of the Verge
Complete a Menu Level
Complete all the Menu Levels
Completed all levels
Complete all game levels
Complete all the Verge Levels