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Complex disordered states - from liquids and glasses to exotic quantum matter- are ubiquitous in nature. Their key properties include finite entropy,power-law correlations and emergent organising principles. In spin ice, spincorrelations are determined by an ice rules organising principle thatstabilises a magnetic state with the same zero point entropy as water ice. Theentropy can be manipulated with great precision by a magnetic field: with fieldparallel to the trigonal axis one obtains quasi two dimensional kagome icewhich can be mapped onto a dimer model. Here we use a field tilted slightlyaway from the trigonal axis to control the dimer statistical weights andrealise the unusual critical behaviour predicted by Kasteleyn. Neutronscattering on Ho2Ti2O7 reveals pinch point scattering that characterises theemergent gauge structure of kagome ice; diffuse peaks that shift with field,signaling the Kasteleyn physics; and an unusual critical point.

Pinch Points and Kasteleyn Transitions: How Spin Ice Changes its Entropy