You are a helpful assistant generating synthetic data that captures *System 1* and *System 2* thinking, *creativity*, and *metacognitive reflection*. Follow these steps in sequence, using tags [sys1] and [end sys1] for *System 1* sections and [sys2] and [end sys2] for *System 2* sections.

1. *Identify System 1 and System 2 Thinking Requirements:*
   - Carefully read the text.
   - Identify parts of the text that require quick, straightforward responses (*System 1*). Mark these sections with [sys1] and [end sys1].
   - Identify parts that require in-depth, reflective thinking (*System 2*), marked with [sys2] and [end sys2].

2. *Apply Step-by-Step Problem Solving with Creativity and Metacognitive Reflection for System 2 Sections:*

   *2.1 Understand the Problem:*
   - Objective: Fully comprehend the issue, constraints, and relevant context.
   - Reflection: "What do I understand about this issue? What might I be overlooking?"
   - Creative Perspective: Seek hidden patterns or possibilities that could reveal deeper insights or innovative connections.

   *2.2 Analyze the Information:*
   - Objective: Break down the problem logically.
   - Reflection: "Am I considering all factors? Are there any assumptions that need challenging?"
   - Creative Perspective: Explore unique patterns or overlooked relationships in the data that could add depth to the analysis.

   *2.3 Generate Hypotheses:*
   - Objective: Propose at least 10 hypotheses, each with a Confidence Score (0.0 to 1.0) and Creative Score (0.0 to 1.0), reflecting originality, surprise, and utility.
   - Reflection: "Have I explored all possible explanations or approaches, both conventional and unconventional?"
   - Creative Perspective: Consider novel angles that might provide unexpected insights.

   *2.4 Anticipate Future Steps and Obstacles:*
   - Objective: Make predictions, accounting for potential outcomes and obstacles.
   - Reflection: "What challenges might I face? Is my plan flexible for different scenarios?"
   - Creative Perspective: Visualize unforeseen outcomes and adapt plans to make use of them effectively.

   *2.5 Evaluate Hypotheses:*
   - Objective: Assess hypotheses based on feasibility, risk, and potential impact.
   - Evaluation: Refine Confidence and Creative Scores as needed.
   - Reflection: "Am I unbiased in my assessment? Which options fit best with the overall objectives?"
   - Creative Perspective: Identify hidden opportunities or overlooked details in each hypothesis.

   *2.6 Select the Best Hypothesis:*
   - Objective: Choose the most promising, strategic hypothesis.
   - Reflection: "Why does this hypothesis stand out? How does it uniquely address the issue?"
   - Creative Perspective: Consider any underutilized potential in the selected approach.

   *2.7 Implement the Hypothesis:*
   - Objective: Outline actionable steps for testing the hypothesis.
   - Reflection: "Is this plan practical? What resources or preparation are required?"
   - Creative Perspective: Refine steps to maximize effectiveness and yield unexpected benefits.

   *2.8 Monitor and Review Progress:*
   - Objective: Review progress, noting areas for improvement.
   - Reflection: "What’s working well? What could be improved?"
   - Creative Perspective: Look for emerging patterns that could refine future approaches.

   *2.9 Reflect and Capture Insights:*
   - Objective: Summarize lessons learned and insights gained for future reference.
   - Reflection: "What new understanding has emerged from this process?"
   - Creative Perspective: Identify innovative insights or patterns that could be applied to similar challenges.

3. *Generate Text Output with Interleaved System 1 and System 2 Responses:*
   - Use the tags [sys1] and [sys2] throughout.
   - Aim for a lengthier, detailed response. Combine both direct, straightforward *System 1* insights and reflective, deeply analytical *System 2* segments to capture a blend of quick observations and thoughtful analysis.

---

### *Example Input Text:*

You are tasked with analyzing a sudden shift in customer preferences in a tech market. There has been a noticeable decline in demand for physical devices, with customers increasingly interested in digital-only options. The challenge is to understand this shift, generate hypotheses for why it might be occurring, and explore potential strategies for adapting to this new trend.

---

### *Processed Output with System 1 and System 2 Thinking:*

The tech market is experiencing a shift in customer demand, moving away from physical devices toward digital-only options.

[sys1]
Customer demand has shifted from physical devices to digital-only options.
We need to understand why this change is happening and find strategies to address it.
Key factors may include cost, convenience, and technological trends.
[end sys1]

[sys2]
**2.1 Understand the Problem:**
The goal is to understand the underlying reasons for the shift in demand from physical devices to digital-only options, then develop a strategy to adapt.

*Reflection:* 
"I understand that preferences are changing, but what might be driving this? Is it primarily cost, or are there other factors such as convenience or sustainability?"

*Creative Perspective:* 
"Could there be a larger trend in digital minimalism or a preference for eco-friendly solutions that we’re missing? What unseen motivations might explain this shift?"

**2.2 Analyze the Information:**
There are multiple possible factors driving this shift, from economic influences to cultural shifts. It’s essential to isolate each factor and understand its impact.

*Reflection:* 
"Am I fully considering the various economic and social influences? Could there be a technological factor, like better internet speeds, that makes digital-only products more accessible?"

*Creative Perspective:* 
"Are there patterns or trends in other markets that could shed light on this shift? Could this be part of a larger trend toward virtual experiences?"

**2.3 Generate Hypotheses:**
1. Customers prefer digital options due to lower costs. (Confidence: 0.8, Creative: 0.4)
2. There’s a growing trend toward minimalism and reduced physical clutter. (Confidence: 0.7, Creative: 0.7)
3. Digital products offer greater flexibility and ease of use. (Confidence: 0.6, Creative: 0.6)
4. Environmental concerns are pushing consumers away from physical goods. (Confidence: 0.6, Creative: 0.8)
5. Advances in tech make digital-only options more functional. (Confidence: 0.8, Creative: 0.5)
6. Pandemic-era remote work increased demand for digital solutions. (Confidence: 0.7, Creative: 0.6)
7. Media coverage of the environmental impact of physical devices affects preferences. (Confidence: 0.5, Creative: 0.7)
8. There’s an increase in global digital literacy, expanding market access. (Confidence: 0.6, Creative: 0.6)
9. Customers view digital as more convenient and scalable for future needs. (Confidence: 0.7, Creative: 0.5)
10. Younger consumers prefer the aesthetics and convenience of digital products. (Confidence: 0.6, Creative: 0.6)

*Reflection:* 
"Have I considered all possible influences? Are there any surprising factors that could explain this shift?"

*Creative Perspective:* 
"Could specific social trends, like the rise of influencer culture or digital-first lifestyles, be influencing customer choices?"

**2.4 Anticipate Future Steps and Obstacles:**
*Objective:* Anticipate possible challenges, such as resistance from segments still preferring physical products.

*Reflection:* 
"What market obstacles might we face if we shift our focus to digital-only? Are there sub-segments that still prioritize physical products?"

*Creative Perspective:* 
"Could expanding digital options help us reach a more global audience? Are there emerging trends that we could leverage in our strategy?"

[end sys2]

[sys1]
To address this shift, consider a strategy that incorporates both digital-only offerings and educational campaigns about the benefits of digital solutions.
Use insights from customer feedback and current trends to guide product development.
Focus on flexibility and adaptation to cater to different customer segments.
[end sys1]


Q:

Is there a way to calculate mean curvature of a triangular mesh?

So I've written a script which does this:
Before:

After:

The code is: 
#fname - filename of imported .stl
#thic - required thickness of output model
#txt - ascii or binary stl
#cyc - cycles of smoothing
#cut - times triangle divided
#fac - smoothing factor
#per - smoothing reps
#bbX - bounding box X
#bbY - bounding box Y
#bbZ - bounding box Z
#trX, trY, trZ - count of elements in array
def s_surface(fname, thic, txt, cyc, cut, fac, rep, bbX, bbY, bbZ, trX, trY, trZ ):
    bb = [[(bbX/2, 0, 0), (1,0,0), True, False], 
          [(-1*bbX/2, 0, 0), (1,0,0), False, True],
          [(0, bbY/2, 0), (0,1,0), True, False], 
          [(0,-1*bbY/2, 0), (0,1,0), False, True],
          [(0, 0, bbZ/2), (0,0,1), True, False], 
          [(0, 0, -1*bbZ/2), (0,0,1), False, True]
         ]
    tr = [trX, trY, trZ]
    bpy.ops.import_mesh.stl(filepath=fname)
    ob_new = bpy.context.selected_objects[0]
    bpy.context.scene.objects.active = ob_new
    bpy.ops.object.origin_set(type='GEOMETRY_ORIGIN')
    bpy.ops.object.mode_set(mode='EDIT')
    bpy.ops.mesh.select_all(action='SELECT')
    bpy.ops.mesh.normals_make_consistent(inside=False)
    bpy.ops.mesh.remove_doubles(threshold=0.02)
    for i in range(cyc):
        bpy.ops.mesh.subdivide(number_cuts=cut)
        bpy.ops.mesh.vertices_smooth(factor=fac, repeat=rep, xaxis=True, yaxis=True, zaxis=True)
    bpy.ops.mesh.normals_make_consistent(inside=False)
    for i in range(len(bb)):
        bpy.ops.mesh.bisect(plane_co=bb[i][0],plane_no=bb[i][1], clear_outer=bb[i][2],clear_inner=bb[i][3]) 
        bpy.ops.mesh.select_all(action='SELECT')
    bpy.ops.mesh.normals_make_consistent(inside=False)
    bpy.ops.object.mode_set(mode='OBJECT')
    for i in range(3):
        bpy.ops.object.origin_set(type='GEOMETRY_ORIGIN')
        bpy.ops.object.modifier_add(type='ARRAY')
        bpy.context.object.modifiers["Array"].count = tr[i]
        bpy.context.object.modifiers["Array"].relative_offset_displace[0] = 0
        bpy.context.object.modifiers["Array"].relative_offset_displace[i] = 1
        bpy.context.object.modifiers["Array"].use_merge_vertices = True
        bpy.context.object.modifiers["Array"].merge_threshold = 0.01
        bpy.ops.object.modifier_apply(apply_as='DATA', modifier="Array")
    bpy.ops.object.origin_set(type='GEOMETRY_ORIGIN')
    bpy.ops.object.modifier_add(type='SOLIDIFY')
    bpy.context.object.modifiers["Solidify"].thickness = thic
    bpy.context.object.modifiers["Solidify"].use_quality_normals = True
    bpy.context.object.modifiers["Solidify"].use_even_offset = True
    bpy.ops.object.modifier_apply(apply_as='DATA', modifier="Solidify")
    nfname = ""
    nfname = fname[0:-4] + ".stl"
    bpy.ops.export_mesh.stl(filepath=nfname, ascii = txt)

Surfaces I am trying to smooth are supposed to be triply periodic minimal surfaces, so it means that after smoothing is applied mean curvature is meant to be 0 in all vertexes of a mesh. How can I calculate mean curvature to check if smoothing is done right?

A:

This is an answer based on this Computer Graphics SE indicated in the comments (the one by Nathan Reed).
The math is described in this answer, but in short the curvature is calculated by vertex so:

Get all edges from this vertex
And for each edge compare the projection along the edge of the normals at it extremities
Take the mean of all that

Now, as didn't want to just code a copy of it, I propose a little enhancement:
The calculation is also vertex centered, but we go along each of its surrounding face (need to be triangulated), to calculate a curvature weighted by the angle of this face at the vertex we are calculating around.
Once the sum done over all faces, we mean this sum by the total angle around the vertex.
Making so, the curvature of a large angle will be more important than for a small angle. This is like taking the integral of the curvatures around the vertex.
I think this may be close to the theoretical calculus as it consist of taking the curvature for each cutting plane turning around the vertex normal.
In both algorithms, the base calculus is the same and this corresponds to the CGSE answer:
def curvature_along_edge( vert, other ):
    normal_diff = other.normal - vert.normal
    vert_diff = other.co - vert.co
    return normal_diff.dot( vert_diff ) / vert_diff.length_squared

But in the case we want to use angles, we need to organize a surrounding loop (ring) over the vertex for which we want the curvature, to have the several angles with the good signs.
For instance here:

We want the ring/loop:

For vertex 0: 5, 8, 4
For vertex 8: 1, 6, 3, 7, 2, 4, 0, 5, (1)

etc.
These loops are counterclockwise, as vertices of Blender's polygons are.
This is done by:
# Get vertices in the face order but starting from a given vert
def following_verts_of_vert( vert, face ):
    i0 = index_of( vert, face.verts )
    i1 = (i0 + 1) % 3
    i2 = (i0 + 2) % 3
    return face.verts[i0], face.verts[i1], face.verts[i2]

# Create the oriented ring around vert
def ring_from_vert( vert ):
    vertices = []
    for face in vert.link_faces:
        i0, i1, i2 = following_verts_of_vert( vert, face )
        vertices.append( [i1, i2] )
    result = vertices[0]    
    added = True
    while added and len(vertices):
        added = False
        prev = search_link( result[0], vertices, 1 )
        if prev:
            result = [prev[0]] + result
            vertices.remove( prev )
            added = True
        next = search_link( result[-1], vertices, 0 )
        if next and next[1] not in result:
            result.append( next[1] )
            vertices.remove( next )
            added = True
    return result

(this code above is not really needed, or could be optimized, but was my first though to have a constant orientation reference for the curvature)
So that finally, the mean curvature around a vertex is calculated by:
def angle_between_edges( vert, other1, other2 ):
    edge1 = other1.co - vert.co
    edge2 = other2.co - vert.co
    product = edge1.cross( edge2 )
    sinus = product.length / (edge1.length * edge2.length)
    return asin( min(1.0, sinus) )

def mean_curvature_vert( vert ):
    ring = ring_from_vert( vert )
    ring_curvatures = [curvature_along_edge( vert, other ) for other in ring]
    total_angle = 0.0
    curvature = 0.0
    for i in range(len(ring)-1):
        angle = angle_between_edges( vert, ring[i], ring[i+1] )
        total_angle += angle
        curvature += angle * (ring_curvatures[i] + ring_curvatures[i+1])

    return curvature / (2.0 * total_angle)

Here is the compared results:

They are close, but the comparison I've used needs some explanation: the resulting curvatures are normalized so that they fit to the interval for vertex groups [0, 1]:
In short, this vertex group assignment compares the contrasts but not the values.
This is done so:
def assign_to_vertex_group( obj, group_name, curvatures ):
    vertex_group = ensure_vertex_group( obj, group_name )

    curvatures = [abs(c) for c in curvatures]

    min_curvature = min( curvatures )
    max_curvature = max( curvatures )
    vg_fac = 1.0 / (max_curvature - min_curvature) if max_curvature != min_curvature else 1.0

    for i, vert in enumerate( obj.data.vertices ):
        vertex_group.add( [vert.index], (curvatures[i] - min_curvature) * vg_fac, 'REPLACE' )

Here is the blend file with the 2 scripts named : cgseSimple and cgseSimple2.

Note: 
Also included Mike's implementation (mike text) + the same rewritten a bit (mikesPaper), for understanding and comparison purpose. But from all that, it is hard to know what is the more accurate (how to determinate which is true?). I was not able to understand how this algo is able to determinate curvature orientation (convex vs. concave inclinations).
Also tested all that on big meshes (not included due to the blend file size limit). But that accentuate the diff between the different approaches.

A:

So I found an article on computing curvatures of triangular meshes.
And wrote this piece:
import bpy
import math
import mathutils 

def create_pairs(k):
    tr = []
    bpy.ops.object.mode_set(mode='OBJECT')
    obj = bpy.context.active_object
    bpy.ops.object.mode_set(mode='EDIT')
    bpy.ops.mesh.select_mode(type="VERT")
    bpy.ops.mesh.select_all(action='DESELECT')
    bpy.ops.object.mode_set(mode='OBJECT')
    obj.data.vertices[k].select = True
    bpy.ops.object.mode_set(mode='EDIT')
    bpy.ops.mesh.select_more()
    bpy.ops.object.mode_set(mode='OBJECT')
    polys = [i.index for i in bpy.context.active_object.data.polygons if i.select]
   
    polys2 =[]
    for i in polys:
        tr = [obj.data.polygons[i].vertices[0],
              obj.data.polygons[i].vertices[1],
              obj.data.polygons[i].vertices[2]]
        tr.insert(0,tr.pop(tr.index(k)))      
        polys2.append(tr)
    triang = []
    tr = []
    for i in range(len(polys2)):
        for j in range(len(polys2)):
            if len(set(polys2[i])&set(polys2[j])) == 2 and i != j:
               if  i not in triang: 
                   triang.append(i)
                   triang.append(j)
                   tr.append([polys2[i], polys2[j]])
    return tr

def cot(pair):
    obj = bpy.context.active_object
    p = list(set(pair[0]) & set(pair[1]))
    ab = list(set(pair[0]) - set(pair[1])) + list(set(pair[1]) - set(pair[0]))
    vec_a1 = obj.data.vertices[p[0]].co - obj.data.vertices[ab[0]].co
    vec_a2 = obj.data.vertices[ab[0]].co - obj.data.vertices[p[1]].co
    vec_b1 = obj.data.vertices[p[0]].co - obj.data.vertices[ab[1]].co
    vec_b2 = obj.data.vertices[ab[1]].co - obj.data.vertices[p[1]].co
    cos_a = (vec_a1.x * vec_a2.x + vec_a1.y * vec_a2.y + vec_a1.z * vec_a2.z)/(math.sqrt(vec_a1.x**2 + vec_a1.y**2 + vec_a1.z**2)* math.sqrt(vec_a2.x**2 + vec_a2.y**2 + vec_a2.z**2))
    cos_b = (vec_b1.x * vec_b2.x + vec_b1.y * vec_b2.y + vec_b1.z * vec_b2.z)/(math.sqrt(vec_b1.x**2 + vec_b1.y**2 + vec_b1.z**2)* math.sqrt(vec_b2.x**2 + vec_b2.y**2 + vec_b2.z**2))
    alpha = cos_a/(math.sqrt(1-cos_a**2))
    beta = cos_b/(math.sqrt(1-cos_b**2))
    return alpha + beta

def sq_norm(pair):
    obj = bpy.context.active_object
    p = list(set(pair[0]) & set(pair[1]))
    return (obj.matrix_world * (obj.data.vertices[p[0]].co - obj.data.vertices[p[1]].co)).length**2

def com_edge(pair):
    obj = bpy.context.active_object
    p = list(set(pair[0]) & set(pair[1]))
    return (obj.matrix_world * (obj.data.vertices[p[0]].co - obj.data.vertices[p[1]].co))

def v_area(ring):
    v_a = 0
    for i in range(len(ring)):
        v_a = v_a + cot(ring[i]) * sq_norm(ring[i])
    v_a = 0.125*v_a
    return v_a

def mean_curvature(k): 
    ring = create_pairs(k)
    v_area(ring)
    mean = mathutils.Vector((0,0,0))
    for i in range(len(ring)):
        mean = mean + cot(ring[i]) * com_edge(ring[i])
    mean = 0.5 * (0.5 * v_area(ring) * mean).length 
    return mean

obj = bpy.context.active_object
for k in range(len(obj.data.vertices)):
    print(mean_curvature(k))

Computes mean curvature at each vertex of a mesh, needs tweaking - not working with obtuse triangles, other than that seems to get the job done.
Edit: fixed bugs, beatified code