Kagome bands disguised in a coloring-triangle lattice

The kagome bands hosting exotic quantum phases generally and understandably pertain only to a kagome lattice. This has severely hampered the research of kagome physics due to the lack of real kagome-lattice materials. Interestingly, we discover that a coloring-triangle (CT) lattice, named after color-triangle tiling, also hosts kagome bands. We demonstrate first theoretically the equivalency between the kagome and CT lattices, and then computationally in photonic (waveguide lattice) and electronic (Au overlayer on electride $\mathrm{C}{\mathrm{a}}_{2}\mathrm{N}$ surface) systems by first-principles calculations. The theory can be generalized to even distorted kagome and CT lattices to exhibit ideal kagome bands. Our findings open an avenue to explore the alluding kagome physics.