# What Do You Know About Maekawa's Theorem?

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In mathematics, Maekawa's theorem is a theorem used in the mathematics of paper folding. It basically involves flat-foldable origami crease patterns that are determined by a variation of valley-creases and mountain-creases.
Maekawa's theorem states that the numbers of mountain folds and valleys always differ by two at every vertex—no matter the direction.

• 1.

### Which aspect of mathematics is Maekawa theorem relevant.

• A.

Mathematics of setting cubes

• B.

Mathematics of folding paper

• C.

Mathematics of wave movement

• D.

Mathematics of scalar surfaces

B. Mathematics of folding paper
Explanation
Maekawa's theorem is relevant to the mathematics of folding paper. This theorem states that when folding a flat sheet of paper along a series of creases, the sum of the angles formed at any vertex will always be a constant, regardless of the number of creases or the specific folding pattern. This theorem is essential in origami and other paper-folding techniques, as it helps determine the feasibility and stability of different folding designs.

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• 2.

### What does Maekawa's theorem propose that the total number of folds at each vertex must be?

• A.

An odd number

• B.

A decimal number

• C.

An even number

• D.

Half of the angle

C. An even number
Explanation
Maekawa's theorem proposes that the total number of folds at each vertex must be an even number. This means that when folding a paper, the number of creases meeting at a vertex should be divisible by 2. This theorem is important in origami and paper folding as it helps ensure that the paper can be folded properly and without any overlapping or intersecting folds.

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• 3.

### Who discovered this theorem before Maekawa worked on it?

• A.

S. Murata

• B.

B. Jurtala

• C.

Jan Nelson

• D.

Robert Oneil

A. S. Murata
• 4.

### What is one major flaw of this theorem?

• A.

It does not completely characterize the flat-foldable vertices

• B.

It shows only the angles of the vertical cortex

• C.

It is only applicable to folds greater than 180 degrees

• D.

The theorem is mutually exclusive to angles of a straight shape

A. It does not completely characterize the flat-foldable vertices
Explanation
The major flaw of this theorem is that it does not provide a complete characterization of the flat-foldable vertices. This means that there are certain types of vertices that can be flat-folded, but are not accounted for in this theorem. Therefore, the theorem is limited in its ability to fully describe and analyze the flat-foldable vertices in a given context.

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• 5.

### When utilizing Maekawa's theorem, which one of the following is applicable at every vertex?

• A.

The number of mountain folds is the same in either direction

• B.

The number of valley and mountain folds is always the same in either direction

• C.

The numbers of valley and mountain folds always differ by two in either direction

• D.

The number of valley fold is the same in both directions of the vertex

C. The numbers of valley and mountain folds always differ by two in either direction
Explanation
According to Maekawa's theorem, the numbers of valley and mountain folds always differ by two in either direction at every vertex. This means that if there are x valley folds in one direction, there will be x+2 mountain folds in that direction, and vice versa. This theorem is a fundamental principle in origami design and ensures that the folds can be properly balanced and the paper can be folded and unfolded correctly.

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• 6.

### Which of the following is applicable when using Maekawa's theorem for a flat-foldable crease pattern?

• A.

It is not possible to color the regions of the creases

• B.

It is always possible to color the regions between the creases with two colors, such that each crease separates regions of differing colors

• C.

Each crease seperates the regions into 5 different colours

• D.

It is not possible to color the region creases with two colors but each crease can still be separated to regions of differing colors

B. It is always possible to color the regions between the creases with two colors, such that each crease separates regions of differing colors
Explanation
Maekawa's theorem states that for any flat-foldable crease pattern, it is always possible to color the regions between the creases with two colors in such a way that each crease separates regions of differing colors. This means that no two adjacent regions on either side of a crease will have the same color. This property is important in origami design as it helps ensure that the folded model can be manipulated without any overlapping or intersecting of the regions.

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• 7.

### Which of these theorems is related to Maekawa's theorem?

• A.

Origami Theorem

• B.

Kawasaki Theorem

• C.

Murata Theorem

• D.

Vannelson Theorem

B. Kawasaki Theorem
Explanation
Kawasaki Theorem is related to Maekawa's theorem. Maekawa's theorem states that the sum of the absolute values of the curvatures of any closed curve on a surface is equal to 2π. Kawasaki's theorem, on the other hand, is a result in origami geometry that states that if a crease pattern satisfies certain conditions, then the resulting folded model will have a flat fold. Both theorems are related to the field of origami and involve geometric properties of folded objects.

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• 8.

### Which of these scientists used the theorem with the same results in the same year it was discovered?

• A.

Jacques Justin

• B.

Hull Thomas

• C.

Takahama Inoshi

• D.

Craig Edwards

A. Jacques Justin
Explanation
Jacques Justin used the theorem with the same results in the same year it was discovered.

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• 9.

### In which mathematical publication was Maekawa's theorem first proposed?

• A.

Theorem of Wide Angles

• B.

Viva Origami

• C.

Theories of Origami Design

• D.

Mathematical Modeling Based on Origami Design

B. Viva Origami
Explanation
Maekawa's theorem is a fundamental result in origami design that relates the number of mountain and valley folds around a vertex. The publication "Viva Origami" is likely to be the source where Maekawa's theorem was first proposed, as it specifically focuses on origami and its mathematical aspects. The other options do not seem to be directly related to origami or mathematical publications.

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• 10.

### Which mathematical problem prompted Jun Maekawa to design the Maekawa's theorem?

• A.

Angle wideness of origami models

• B.

Crease regions in origami models

• C.

Fold number in origami models

• D.

Flat-foldability of origami models

D. Flat-foldability of origami models
Explanation
Jun Maekawa designed Maekawa's theorem to address the mathematical problem of flat-foldability in origami models. This theorem explores the conditions under which a two-dimensional sheet of paper can be folded into a three-dimensional shape without any overlaps or creases. By studying the flat-foldability of origami models, Maekawa aimed to understand the principles and limitations of folding paper and develop new design techniques for origami artists.

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