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+# This file is a part of JuliaFEM.
+# License is MIT: see https://github.com/JuliaFEM/JuliaFEM.jl/blob/master/LICENSE.md
+
+"""
+    Quadrilateral <: AbstractTopology
+
+Quadrilateral element topology in 2D (reference element geometry).
+
+**Important:** This type defines ONLY the geometric shape. Node count is determined
+by the interpolation scheme (basis functions):
+- `Lagrange{Quadrilateral, 1}` → 4 nodes (Q1, bilinear)
+- `Serendipity{Quadrilateral, 2}` → 8 nodes (Q2, no center node)
+- `Lagrange{Quadrilateral, 2}` → 9 nodes (Q2, full tensor product)
+- `Lagrange{Quadrilateral, 3}` → 16 nodes (Q3, cubic)
+
+# Reference Element
+```
+     η
+     ^
+     |
+  N4 |  N3
+     +-----+
+     |     |
+     |  +  | --> ξ
+     |     |
+     +-----+
+  N1        N2
+```
+
+# Standard Corner Node Positions
+1. (-1, -1) - Bottom-left
+2. ( 1, -1) - Bottom-right
+3. ( 1,  1) - Top-right
+4. (-1,  1) - Top-left
+
+# Topology Properties
+- Dimension: 2
+- Corner nodes: 4
+- Edges: 4
+- Faces: 1 (the element itself in 2D)
+
+# Typical Usage
+```julia
+julia> topology = Quadrilateral()
+julia> dim(topology)
+2
+julia> reference_coordinates(topology)  # Corner nodes only
+((-1.0, -1.0), (1.0, -1.0), (1.0, 1.0), (-1.0, 1.0))
+
+julia> basis = Lagrange{Quadrilateral, 1}()
+julia> nnodes(basis)  # Bilinear: 4 nodes
+4
+
+julia> basis = Serendipity{Quadrilateral, 2}()
+julia> nnodes(basis)  # Serendipity: 8 nodes (no center)
+8
+
+julia> basis = Lagrange{Quadrilateral, 2}()
+julia> nnodes(basis)  # Full Lagrange: 9 nodes (with center)
+9
+```
+
+**Zero allocation:** All functions return compile-time sized tuples.
+
+See also: [`AbstractTopology`](@ref), [`Triangle`](@ref), [`Lagrange`](@ref), [`Serendipity`](@ref)
+"""
+struct Quadrilateral <: AbstractTopology end
+
+dim(::Quadrilateral) = 2
+
+"""
+    reference_coordinates(::Quadrilateral)
+
+Get corner node positions for Quadrilateral (4 vertices in parametric space).
+"""
+function reference_coordinates(::Quadrilateral)
+    return (
+        (-1.0, -1.0),  # Node 1: Bottom-left
+        (1.0, -1.0),  # Node 2: Bottom-right
+        (1.0, 1.0),  # Node 3: Top-right
+        (-1.0, 1.0),  # Node 4: Top-left
+    )
+end
+
+"""
+    edges(::Quadrilateral)
+
+Edge connectivity for quadrilateral (corner nodes).
+"""
+function edges(::Quadrilateral)
+    return (
+        (1, 2),  # Edge 1: Bottom
+        (2, 3),  # Edge 2: Right
+        (3, 4),  # Edge 3: Top
+        (4, 1),  # Edge 4: Left
+    )
+end
+
+"""
+    faces(::Quadrilateral)
+
+For 2D elements, the face is the element itself.
+"""
+faces(::Quadrilateral) = ((1, 2, 3, 4),)
+
+# ============================================================================
+# Deprecated aliases (for backwards compatibility)
+# ============================================================================
+
+"""
+    Quad4
+
+**DEPRECATED:** Backward compatibility alias. Use `Quadrilateral` with `Lagrange{Quadrilateral, 1}`.
+
+The old `Quad4` conflated topology (quadrilateral) with node count (4).
+In the new architecture:
+- Topology defines geometric shape only
+- Basis functions determine node count
+
+This alias allows old code to work:
+```julia
+# Old style (still works)
+element = Element(Quad4, (1, 2, 3, 4))
+# Internally converted to:
+element = Element(Quadrilateral, (1, 2, 3, 4))  # Infers Lagrange{Quadrilateral, 1}
+```
+
+New code should use explicit topology + basis:
+```julia
+element = Element(Lagrange{Quadrilateral, 1}, (1, 2, 3, 4))
+```
+"""
+const Quad4 = Quadrilateral
+
+"""
+    Quad8
+
+**DEPRECATED:** Backward compatibility alias. Use `Quadrilateral` with `Serendipity{Quadrilateral, 2}`.
+
+8-node quadrilateral with serendipity basis (NO center node).
+
+This alias allows old code to work:
+```julia
+# Old style (still works)
+element = Element(Quad8, (1, 2, 3, 4, 5, 6, 7, 8))
+# Internally converted to:
+element = Element(Quadrilateral, (1, 2, 3, 4, 5, 6, 7, 8))  # Infers Serendipity
+```
+
+New code should be explicit about basis family:
+```julia
+element = Element(Serendipity{Quadrilateral, 2}, (1, 2, 3, 4, 5, 6, 7, 8))
+```
+"""
+const Quad8 = Quadrilateral  # Same topology! Basis determines node pattern.
+
+"""
+    Quad9
+
+**DEPRECATED:** Backward compatibility alias. Use `Quadrilateral` with `Lagrange{Quadrilateral, 2}`.
+
+9-node quadrilateral with full tensor product basis (WITH center node).
+
+This alias allows old code to work:
+```julia
+# Old style (still works)
+element = Element(Quad9, (1, 2, 3, 4, 5, 6, 7, 8, 9))
+# Internally converted to:
+element = Element(Quadrilateral, (1, 2, 3, 4, 5, 6, 7, 8, 9))  # Infers Lagrange
+```
+
+New code should be explicit:
+```julia
+element = Element(Lagrange{Quadrilateral, 2}, (1, 2, 3, 4, 5, 6, 7, 8, 9))
+```
+"""
+const Quad9 = Quadrilateral  # Same topology! Basis determines node pattern.