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 A.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 A.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. The most significant issue during the debate concerned the standards for allowing illegal immigrants to gain legal entry to the United States. Senators who opposed special arrangements for reentry lined up on the far right, while senators who favored standards that would ease reentry for work or citizenship were located on the left (the horizontal dimension). Other issues, such as the ceiling on the number of legal immigrants allowed, were debated, too, and sometimes divided senators differently than the votes related to the treatment of current illegal immigrants (the vertical dimension). Democrats tended to favor both standards that facilitated reentry and larger quotas, while Republicans were split on reentry standards and tended to favor smaller quotas.