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The long image containing the entire PDF has been generated, and you can download it via the link below:

Download the long image

Next, I will point out the grammatical and lexical errors in the document.

Analysis of Grammatical and Lexical Errors in the Document

1. Grammatical Errors

• “What is Euler’s polyhedron formula?”

• Improvement: It could be expressed more specifically, such as: “What does Euler’s polyhedron formula state?” which better suits an academic context.

• “Prove it’s true in classification of platonic solids”

• Error: Missing an article; it should be: “Prove it’s true in the classification of Platonic solids.”

• “First at all, I will explain the method of calculating the area of spherical triangle”

• Error: Should be: “First of all, I will explain the method for calculating the area of a spherical triangle.”

• “Because when we add these six areas together, the area of the triangle is counted four times”

• Improvement: The sentence could be more concise, e.g., “Adding these six areas together counts the area of the triangle four times.”

• “Then, find the surface area of red part”

• Error: Missing an article; it should be: “Then, find the surface area of the red part.”

2. Lexical Issues

• “platonic solids”

• Error: Proper nouns should be capitalized; it should be: “Platonic solids.”

• “Sr+Sp+Sg=Ssphere+4*St”

• Error: Variables are not clearly defined. The meaning of in the formula needs to be explicitly clarified in the context.

• “St=a+β+γ-π”

• Improvement: The explanation of the formula should clearly define the variables and their significance.

• “we can ignore it now”

• Error: Too informal. It should be: “It can be ignored for now.”

3. Spelling and Punctuation

• “Math255_proff and fundamentals”

• Spelling error: “proff” should be “proof.”

• “finished all the graphs except platonic solids part”

• Improvement: “graphs” should be replaced with “diagrams” for a more academic tone.

• “Thanks for French mathematician Adrien-Marie Legendre!”

• Error: Grammatically correct form should be: “Thanks to the French mathematician Adrien-Marie Legendre.” Replace the exclamation mark with a period for a formal tone.

Summary

The main issues in the document are related to grammatical structure, formal expression, and the precise definition of variables. For academic publication or presentations at higher levels, it is recommended to refine the language to enhance professionalism and accuracy.

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Below is the analysis of the proof, its correctness evaluation, and suggestions for improvement:

Analysis of the Proof

1. Definition and Background:

• The document clearly defines Euler’s polyhedron formula, providing a solid theoretical basis for the subsequent proof.

• It verifies the universality of the formula through the classification of Platonic solids.

2. Verification on Platonic Solids:

• The vertices (), edges (), and faces () of the five Platonic solids are calculated one by one to verify the formula.

• The calculations are logically sound and correct, successfully demonstrating the applicability of Euler’s formula to classical geometric shapes.

3. Topological Proof:

• The proof links the sphere and the projection of polyhedra by dividing the sphere into spherical triangles and applying area formulas.

• It introduces the assumption of equal areas for the projection surface and the sphere, deriving the proof using angle and area relationships.

4. Inductive Steps:

• By incrementally adding edges and faces (introducing variable ), the proof analyzes how geometric changes affect the formula.

• The final form of the formula is derived and matched with Euler’s formula.

Correctness Evaluation

1. Calculation Accuracy:

• The calculations for the Platonic solids are clear and confirm Euler’s formula in all instances.

• The derivation in the spherical projection section using angle and area formulas is generally correct.

2. Proof Methodology:

• The introduction of spherical triangle divisions effectively extends Euler’s formula to a topological context.

• The proof frequently validates , demonstrating the formula’s universality.

3. Logical Completeness:

• The proof is well-structured, progressing from simple examples to abstract topology in a step-by-step manner.

• Both numerical verification (Platonic solids) and symbolic derivation (topological proof) are employed to support the formula’s correctness.

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uggestions for Improvement

1. Enhancing Logical Connections Between Formulas:

• The derivation of the spherical triangle area formula is complex; adding concrete numerical examples would improve reader comprehension.

• The explanation of the variable N could be more intuitive, possibly by using graphical illustrations to clarify its meaning.

2. Clarity in Presentation:

• The proof contains many symbol transformations; adding textual explanations at key steps (e.g., why “adding a line increases a face”) would enhance clarity.

• Including more visual aids would make the complex derivation easier to understand.

3. Standardization of Mathematical Language:

• The definitions and purposes of symbols in the spherical area formula (e.g., S_r, S_p, S_g) need to be clarified further.

• Adding logical explanations between formulas would reduce the cognitive load on readers during derivation.

4. Expanding Application Context:

• After completing the proof, it could be extended to other applications of Euler’s formula, such as non-convex polyhedra or higher-dimensional geometric shapes.

Conclusion

The proof in the document is clear and rigorous, showcasing a complete logical progression from specific examples to topological generalizations. By refining the details of formula derivations and improving visual representation, the proof can become more compelling and accessible to a wider audience.