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For continuity across the mesh, the vertex count on the combined edges should be equal. In this example each edge has three vertex points.
What Just Happened?
To see what happens when two objects are combined and merged, turn on the Heads Up Display and note the vertex and edge count. To activate the Heads Up Display, choose Display > Heads Up Display > Poly Count.
In this example, prior to merging, there were 18 vertices and 24 edges. After merging, the count becomes 15 vertices and 22 edges. The center vertices and edges merge.
It is often handy to merge two separate polygon objects into a single polygon mesh.
Move polygon objects close to each other. Shift-click to select both objects and combine them into one object (Mesh > Combine) from the Polygons menu set.
With the object selected, click the Component, Vertices icon.
Align the points and edges that will be joined together in Component Mode. Use the Snap to Point feature (v key) to snap to point while moving points.
Return to the Object mode, select the object and choose Edit Mesh > Merge from the Polygons menu set. Set Threshold Distance in options if necessary. The Threshold Distance sets the maximum distance that two vertices can be apart to be merged.