Immunogen design approaches aim to control the specificity and quality of antibody responses to enable the creation of next-generation vaccines with improved potency and breadth. However, our understanding of the relationship between immunogen structure and immunogenicity is limited. Here we use computational protein design to generate a self-assembling nanoparticle vaccine platform based on the head domain of influenza hemagglutinin (HA) that enables precise control of antigen conformation, flexibility, and spacing on the nanoparticle exterior. Domain-based HA head antigens were presented either as monomers or in a native-like closed trimeric conformation that prevents exposure of trimer interface epitopes. These antigens were connected to the underlying nanoparticle by a rigid linker that was modularly extended to precisely control antigen spacing. We found that nanoparticle immunogens with decreased spacing between closed trimeric head antigens elicited antibodies with improved hemagglutination inhibition (HAI) and neutralization potency as well as binding breadth across diverse HAs within a subtype. Our "trihead" nanoparticle immunogen platform thus enables new 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 to generate next-generation vaccines against influenza and other viruses.