Exercise 2 — The Topology Puzzle, the Muscle, and the Repair

What you're learning

The one move that changes face-loop count and redirects edge flow — then how to use it on a real
joint.

Time: 90 minutes
New tools: Insert Edge Loop, Multi-Cut, Smooth, Extrude

20 minutes, no computer

Copy the patch onto paper: two square faces across the top, three square faces across the
bottom, empty middle, border only.

┌──────┬──────┐
│      │      │
├──────┴──────┤
│             │
│    solve    │
│             │
├─────┬───┬───┤
│     │   │   │
└─────┴───┴───┘

Rules: all faces must be four-sided; do not alter the top or the bottom.
Work alone for five minutes. No hints, no phones.
We compare solutions at the board. Some will be wrong in useful ways — a triangle, a five-sided
face, an altered bottom row.
In each valid solution, find the triple point — the vertex where three edges meet.
Find where a loop exited to the side border.
Name the two operations out loud: a loop carried straight through, and a loop terminated at a
triple point. This is the general solution for changing loop count and redirecting edge flow.

30 minutes

Create a polygon cube, 1 subdivision per axis, 20 units.
Delete the bottom face so the cube is open at the midpoint.
Select the mesh and apply Mesh → Smooth, divisions 2.
Orbit and look at the result: a dome with a clean square boundary ring and dense quads across
the top.
Select the open boundary edge loop. This ring is what stitches into an arm's edge loops.
Count its edges and note that it is a power-of-two multiple of 4 — so it will only stitch
cleanly into an arm with a matching loop count.
Find the four poles. Confirm they sit at the original cube corners, away from the center.
The bulging region is all quads.
Now the second route: create a polygon plane, 3 × 3 faces.
Select the center face. Edit Mesh → Extrude. Scale the extrusion in slightly, then pull it
up.
Look at the topology surrounding the extrusion. It is the puzzle solution — a loop carried
through, and loops terminating at triple points at the corners.

40 minutes

Open week02_elbow_BAD.ma. Count the loops crossing the elbow. Identify which one would be
the holding loop and which would travel with each joint.
Insert loops with Mesh Tools → Insert Edge Loop until the three-loop rule is satisfied.
Space them so the middle loop sits at the crease.
Open week02_shoulder_BAD.ma. The problem here is direction, not count. Use Multi-Cut
and the puzzle solution to redirect the flow into the deltoid without adding loops that run
the whole length of the arm.
Test both: apply a nonlinear Bend deformer (Deform → Nonlinear → Bend) and drive it
through 90°. Judge the crease. Then delete the deformer.
Write two sentences naming which failure was a count problem and which was a direction
problem. You will make this same distinction again in Week 6, when a weighting problem turns
out to be a topology problem.


Puzzle solved correctly, all quads, top and bottom unaltered
Triple point identified
Muscle dome and extruded patch both built
Elbow bends 90° without creasing
Shoulder flow redirected, not just loop-stuffed
All quads — no triangles, no n-gons in any deforming region
Count-vs-direction distinction correctly made in writing