Julie, at https://ispeakmath.org, describes a snowball activity that helps students get to know one another. Throwing snowballs on the first day helps to ease the tension; and speaking aloud what someone else wrote removes the fear of sharing personally.
Give students 3- 5 pipe cleaners. Invite them to create a structure or symbol to represent themselves. Then organize speed dating so that students can share their work!
Get students talking about themselves ... wouldn't have to use Geometry vocabulary ... could use any math word!
Use creating graphing stories for students to share something about themselves. Everything you need is in this blog post!
Dan Meyer's Who I Am has been popular for several years. Ask students to fill it out ... save it ... share it with them at the end of the year!
D. Meyer shares an activity by another teacher (Rich) in which students manipulate a circle to discuss various geometric forms (follow the link). At the end of the activity, each student has a truncated tetrahedron. Students then work together to glue/tape their pieces together to form an icosahedron. Here is a link to the instructions: http://bit.ly/2umDhq5
This or That gives students a chance to find commonalities among them. This site has 350+ this or that choices - not all school appropriate but definitely more than enough to choose from.
Cult of Pedagogy makes a good point that too many icebreakers require social risks too great for a first day/first week or they don't help students get to know one another. Check out the three suggestions for icebreakers on this blog post!