Much of congressional politics has geometric characteristics. When we speak of most Democrats as liberals, most Republicans as conservatives, and some legislators as moderates, we have in mind an ideological or policy spectrum – a line or dimension – along which we can place legislators. In recent Congresses, the parties have been sharply divided, with very little overlap between the parties. Figure 40.1 illustrates this for the 109th Congress (2005–2006) for senators. Using a statistical technique, senators were scored on the basis of their overall voting record in the Congress. Democrats and Republicans were concentrated on opposite sides of the spectrum, creating one of the most polarized Senates in history.

Legislators' policy positions also can be represented in two or more dimensions, when appropriate. In Figure 40.2, senators' policy positions are identified in two dimensions for a debate on an immigration reform bill in 2006. Their locations are identified with the help of a statistical analysis of their votes on about three dozen amendments and other motions that were considered on the Senate floor.