A notable feature of advanced mathematics is that much of it is concerned with geometry in more than three dimensions. ‘Dimension’ looks at high-dimensional geometry, which is best understood from an abstract point of view. Rather than worrying about the existence, or otherwise, of twenty-six dimensional space, let us think about its properties. How is high-dimensional space defined? Can four-dimensional space be visualized? What is the point of higher-dimensional geometry? All of these questions are considered along with fractional dimension, which sheds interesting light on the twin processes of abstraction and generalization.