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The exercise in your photos presents a set of interesting questions about geometry, symmetry, and natural patterns. Here's a summary of the questions along with some guidance on how to solve them:

### 1. **Example of a Natural Pattern Without Numbers**
   - **Answer Idea**: Consider patterns like the **spiral arrangement of leaves** (phyllotaxis), the **branching patterns of trees**, or **crystals** forming in minerals. These patterns occur in nature without explicit reference to numerical values but are deeply mathematical.

### 2. **Golden Ratio Division of a Line Segment**
   - The length \(L = 12\) units needs to be divided into two segments \(L_1\) and \(L_2\) using the golden ratio \( \phi \approx 1.618 \).
   - The rule for dividing a segment into golden sections is:
     \[
     \frac{L_1}{L_2} = \phi
     \]
     Let \(L_1\) be the longer segment. Then, the total length can be expressed as:
     \[
     L_1 + L_2 = 12
     \]
     Using the golden ratio:
     \[
     L_1 = \phi L_2
     \]
     Substituting into the total length equation:
     \[
     \phi L_2 + L_2 = 12
     \]
     Solve for \(L_2\), and then \(L_1\) can be calculated by multiplying \(L_2\) by \( \phi \). The values can then be rounded to two decimal places.

### 3. **Construct a Logarithmic Spiral Using Golden Triangles**
   - A **logarithmic spiral** grows by a constant angle for each turn, and it can be constructed using **golden triangles**, where the ratio of the longer side to the shorter side is the golden ratio. Given that the isosceles side is 3 units, you can create a series of golden triangles where the ratio between consecutive side lengths follows the golden ratio to approximate the spiral.

### 4. **Logarithmic Spiral Using Golden Rectangles**
   - Similar to the previous question, but in this case, **golden rectangles** are used, where the ratio of the length to the width is \( \phi \). Starting with a rectangle of 4 units as a side, you can construct smaller rectangles inside using the golden ratio and then draw quarter circles inside each to form the spiral.

### 5. **Assess Symmetry Using Phi (Golden Ratio)**
   - You can measure certain proportions of your body (e.g., height to the distance from your navel to the floor) and check if they approximate the golden ratio. Similarly, facial measurements like the distance between the eyes compared to the width of the face might show a close resemblance to the golden ratio.

### 6. **Spherical vs. Plane Triangles**
   - **Plane triangles** are what we typically study in Euclidean geometry, with angles summing to \(180^\circ\).
   - **Spherical triangles** exist on the surface of a sphere, and the sum of their angles is greater than \(180^\circ\), depending on the size of the triangle relative to the sphere.

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Would you like me to go deeper into any of these concepts, or solve any specific part in more detail?

### 5 Relative Questions:
1. How do logarithmic spirals appear in nature, such as in seashells or galaxies?
2. Can you explain how the golden ratio appears in famous works of art or architecture?
3. What is the mathematical definition of a spherical triangle, and how is it calculated?
4. What are some real-world applications of the golden ratio in modern design or engineering?
5. How does the golden rectangle relate to Fibonacci numbers?

### Tip:
When working with the golden ratio, it's often helpful to round \(\phi\) to 1.618 for practical purposes, but in precise calculations, use the full value or a calculator for higher accuracy.

The exercise presents various geometry-related problems, such as dividing a line segment using the golden ratio, constructing logarithmic spirals using golden triangles and rectangles, and assessing body symmetry using the golden ratio.

Learn how to divide a line segment using the golden ratio, construct logarithmic spirals with golden triangles and rectangles, and assess symmetry with the golden ratio. Explore the differences between spherical and plane triangles in this high school-level geometry exercise.

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