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To begin, we need to define malleability and ductility which are the mechanical properties of metals. \( \textbf{Malleability:}\) Malleability is the ability of a metal to be hammered, rolled, or pressed into thin sheets without breaking. It is a measure of how easily a metal can be deformed under compressive stress. \( \textbf{Ductility:}\) Ductility is the ability of a metal to be drawn into thin wires without breaking. It is a measure of how easily a metal can be deformed under tensile stress.

Next, we need to understand the structure of metallic bonding. In metallic bonding, metal atoms are surrounded by a "sea" of free electrons. These electrons are delocalized, meaning they are not confined to a single bond or atom but are free to move throughout the entire metal structure, resulting in strong electrostatic attractions between the positively charged metal ions and negatively charged electrons. This creates a unique bonding structure that is responsible for the characteristic properties of metals.

We can now connect the properties of malleability and ductility to metallic bonding. The delocalization of electrons in the "sea" of electrons allows metal ions to slide past one another without breaking the overall metallic bonding. This sliding action enables metals to undergo deformation without breaking, giving them their malleable and ductile properties. When a metal is subjected to compressive stress (as in the case of malleability), the metal ions are forced closer together. The delocalized electrons can still maintain the electrostatic attraction between the ions, allowing the metal to be deformed without breaking. When a metal is subjected to tensile stress (as in the case of ductility), the metal ions are pulled apart, and the "sea" of electrons can still maintain the electrostatic attractions between the ions. This allows the metal to be pulled into thin wires without breaking. In both malleability and ductility, the metallic bonding structure remains intact due to the mobile nature of the delocalized electrons. This ability to maintain strong bonding while allowing deformation under stress is what makes metals malleable and ductile.