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This paper demonstrates the application of a design tool called Bio TRIZ. Its developers claim that it can be used to access biological strategies for solving engineering problems. Our aim is to design a roof for hot climates that gets free cooling through radiant coupling with the sky. The insulation in a standard roof stops the sun and convection from warming the thermal mass.But it also restricts the mass's longwave view of the cool sky. Different solutions to this conflict are offered by BioTRIZ. The chosen solution is to replace the standard insulation component with an open cell honeycomb. The vertical cells would allow longwave radiation to pass, while arresting convection. The solutions offered by BioTRIZ's technological counterpart include no such changes in structure. It is estimated that the thermal mass in the biomimetic roof would remain on average 4.5℃ cooler than in a standard roof over a year in Riyadh, Saudi Arabia.