Immunogen design approaches aim to control the specificity and quality of antibody responses elicited by next-generation vaccines. Here, we use computational protein design to generate a nanoparticle vaccine platform based on the receptor-binding domain (RBD) of influenza hemagglutinin (HA) that enables precise control of antigen conformation and spacing. HA RBDs are presented as either monomers or native-like closed trimers that are connected to the underlying nanoparticle by a rigid linker that is modularly extended to precisely control antigen spacing. Nanoparticle immunogens with decreased spacing between trimeric RBDs elicit antibodies with improved hemagglutination inhibition and neutralization potency as well as binding breadth across diverse H1 HAs. Our "trihead" nanoparticle immunogen platform provides insights into anti-HA immunity, establishes antigen spacing as an important parameter in structure-based vaccine design, and embodies several design features that could be used in next-generation vaccines against influenza and other viruses.