What in the World Are 4D Shapes?

ABSTRACT It is often said that in special relativity, 4D shapes can be used to explain 3D shape projections onto frames of reference. Perdurantists have frequently used this as a motivation for their view of persistence. Despite this, philosophers never clearly state what 4D shapes are. We take a foundational approach toward this issue, first getting clear on the nature of 4D shapes and then using our analysis of 4D shapes to question the explanatory connection between 4D shapes and 3D shapes. We begin by considering some accounts of 4D shapes that seem implicit in philosophers' remarks, finding them deficient. We then discuss what it takes for two objects to have the same 4D shape, working backward to an account of what 4D shapes are. This leads us to argue that 4D shapes should be understood in terms of the isometries of Minkowski spacetime. Subsequently, we use this analysis to show that 4D shapes do not explain 3D projections onto frames in the way that perdurantists have suggested.