Here is the full list of all 26 Square Norm achievements.
Collect 5 Stars.
Listen to title screen music 5 times in a row.
Unlocked the Pentagon shape.
Unlocked the Hexagon shape.
Unlocked the Heptagon shape.
Unlocked the Octogon shape.
Unlocked the Nonagon shape.
Unlocked the Decagon shape.
Unlocked the Undecagon/Hendecagon shape.
Unlocked the Dodecagon shape.
Unlocked Circle Mode
Defeat 100 Irregulars
Defeat 500 Irregulars
Defeat 1000 Irregulars
Defeat 2000 Irregulars
Collect 60 Stars
Collect 30 Stars
Collect 90 Stars
Collect 120 Stars
Collect 150 Stars
Collect 180 Stars
Collect 210 Stars
Collect 240 Stars
Collect All 270 Stars
Defeat the Final Boss
Die 250 times.