There are a number of areas of application in which geometry is a central contributor.

One area that involves geometric constructions, and tessellations is the structure, and the behaviour of crystals.
A look at the wiki pages for crystal structure, symmetries, etc. will show a great deal of geometry, including links to the 2D version - which is tessellations.
For crystals, the current research includes work on the mechanics of when and how crystal structures flex and move. For some crystals, such as Zeolite, this seems to be central to how they are used. Again a check of wiki pages on Zeolites will speak to how ubiquitous these materials are. They occur in kitty litter (absorbing ammonia) and in packaging for fruits and vegetables (to absorb the chemical which causes them to ripen too fast)!

Part of what is going on now involves mathematical modeling of the various crystal structures - there are 230 types, mathematically classified in ways that extend the classification of 17 types of plane tessellations.

Now - to suggest several additional areas of application of geometric constructions:

• robotics: design of robots, motion planning or computing paths ....
  
  
• mechanical engineering - design and analysis of linkages for various purposes;
  
  
• biochemistry of proteins - which are often biological versions of machines, with motions governed by the same principles as mechanical linkages.

All of this areas of current work in geometry.
For a peak at some of the areas of current research, you can look at the 8 weeks of workshops at the Fields Institute this semester:
http://www.fields.utoronto.ca/programs/scientific/11-12/discretegeom/

Walter Whiteley
York University