Three-dimensional lattice modulations in the charge density wave system <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub> <mml:mi>Lu</mml:mi> <mml:mn>2</mml:mn> </mml:msub> <mml:msub> <mml:mi>Ir</mml:mi> <mml:mn>3</mml:m

Using total and resonant x-ray scattering coupled to large-scale computer modeling, we study the lattice modulations in the complex charge density wave (CDW) material ${\mathrm{Lu}}_{2}{\mathrm{Ir}}_{3}{\mathrm{Si}}_{5}$. We find that it is a unique quantum system where periodic lattice modulations related to emergent CDW order occur in three orthogonal atomic planes of the crystal lattice, leading to the emergence of an unusual three-dimensional (3D) pattern of short and long Ir-Ir and Lu-Lu bonds. The 3D character of observed lattice modulations explains the largely isotropic character of the changes in the electronic properties occurring when the CDW order sets in, demonstrating the strong electron-lattice coupling in ${\mathrm{Lu}}_{2}{\mathrm{Ir}}_{3}{\mathrm{Si}}_{5}$. The result is supported by DFT calculations based on the experimental structure data. Altogether, our work provides strong evidence for the presence of a relationship between the dimensionality of emergent lattice distortions and that of concurrent changes in the electronic properties of CDW materials. The relationship may need to be accounted for when these materials are explored for practical applications.