How do you solve complex origami problems using geometric reasoning and logic?

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Origami is the art of folding paper into various shapes and forms, often inspired by nature, animals, or abstract patterns. But did you know that origami can also be a way of exploring geometric concepts, such as angles, polygons, symmetry, and transformations? In this article, you will learn how to use geometric reasoning and logic to solve complex origami problems, such as creating regular polygons, bisecting angles, and dividing regions. You will also discover some of the applications and benefits of origami in mathematics, engineering, and design.

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Abhiram Sripat

COO at Florence Quantum Labs | Innovating Climate Solutions with Quantum Machine Learning

1 Basic folds and axioms

To start with, you need to master some basic folds and axioms that are the foundation of origami geometry. Folds are the lines that you create by bending the paper, and axioms are the rules that describe how to make certain folds. There are seven origami axioms, each of which allows you to fold the paper in a specific way, such as folding a point to a point, folding a point to a line, or folding two points onto two points. You can use these axioms to construct various geometric figures, such as perpendicular lines, bisectors, midpoints, and congruent angles.

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Abhiram Sripat

COO at Florence Quantum Labs | Innovating Climate Solutions with Quantum Machine Learning
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Solving complex origami problems involves applying geometric reasoning and logical steps rooted in basic folds and origami axioms. Start by mastering fundamental folds like the valley, mountain, and reverse folds, which form the foundation for more intricate designs. Use Huzita-Hatori axioms, which define permissible paper folding operations, to create precise angles and intersections. Analyze the desired outcome and decompose it into smaller, manageable sections. Apply geometric principles to ensure accurate alignment and symmetry. By systematically combining these basic folds and axioms, complex origami structures can be methodically and logically constructed, transforming flat paper into intricate, three-dimensional art.

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2 Regular polygons

One of the challenges of origami is to create regular polygons, which are shapes with equal sides and angles, such as triangles, squares, pentagons, and hexagons. To do this, you need to use the concept of angle trisection, which means dividing an angle into three equal parts. For example, to make a regular hexagon, you can start with a square paper and fold one corner to the opposite side, creating a 60-degree angle. Then, you can trisect this angle using axiom 4, which allows you to fold two points onto two points. By repeating this process on the other corners, you will end up with a hexagon.

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Abhiram Sripat

COO at Florence Quantum Labs | Innovating Climate Solutions with Quantum Machine Learning
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Creating regular polygons in origami requires a blend of geometric reasoning and precise folding techniques. Start by folding the paper to identify key reference points and lines that will guide the construction. For instance, to create a regular hexagon, fold the paper to form equilateral triangles, utilizing their internal angles to achieve accurate 60-degree intersections. Huzita-Hatori axioms can be applied to ensure precise bisecting of angles and alignment of edges. By systematically folding and aligning these sections, the paper transforms into a regular polygon, with all sides and angles equal, showcasing the power of geometric principles in origami.

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3 Angle bisectors

Another useful skill in origami is to bisect an angle, which means dividing it into two equal parts. This can help you create symmetrical shapes, such as butterflies, stars, and flowers. To bisect an angle, you can use axiom 2, which allows you to fold a point to a line. For example, to bisect a 90-degree angle, you can fold the paper so that the corner point touches the opposite edge, creating a 45-degree angle. By doing this on both sides of the corner, you will bisect the original angle.

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4 Region division

A more advanced origami problem is to divide a region into equal parts, such as halves, thirds, or quarters. This can help you create complex patterns, such as tessellations, spirals, and fractals. To divide a region, you need to use the concept of geometric mean, which is the square root of the product of two numbers. For example, to divide a square into three equal parts, you can fold one edge to the opposite edge, creating a rectangle. Then, you can fold the shorter side of the rectangle to the longer side, creating a right triangle. The hypotenuse of this triangle is the geometric mean of the shorter and longer sides. By folding the paper along this hypotenuse, you will divide the square into three parts.

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Abhiram Sripat

COO at Florence Quantum Labs | Innovating Climate Solutions with Quantum Machine Learning
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Dividing a region in origami involves creating specific folds to partition the paper into equal or proportional sections. Start by identifying the area you need to divide. Create reference points by folding the paper to form creases that act as guides, such as folding in half to mark the midpoint. For equal divisions, fold one edge to the reference crease and make a new crease, repeating the process for the other sections. Fine-tune the creases to ensure precision, and verify accuracy by folding and aligning the sections. This methodical approach enables accurate region division, facilitating the creation of intricate origami designs.

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5 Applications and benefits

Origami is not only a fun and creative hobby, but also a valuable tool for learning and applying geometry. By folding paper, you can visualize and manipulate geometric concepts, such as angles, polygons, symmetry, and transformations. You can also develop your spatial reasoning, logic, and problem-solving skills, which are essential for many fields of science, technology, engineering, and mathematics. Moreover, origami has inspired many innovations and designs in architecture, robotics, medicine, and art, such as foldable structures, origami robots, stent grafts, and kinetic sculptures.

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Abhiram Sripat

COO at Florence Quantum Labs | Innovating Climate Solutions with Quantum Machine Learning
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Dividing regions in origami has various applications and benefits in both artistic and practical domains. In art, precise region division allows for the creation of complex, symmetrical designs, such as tessellations and modular origami, enhancing the aesthetic appeal. In education, it helps teach geometric concepts, spatial reasoning, and problem-solving skills. Practically, it aids in developing efficient packing and folding techniques, applicable in fields like materials science and robotics. Moreover, mastering region division improves precision and attention to detail, fostering a deeper appreciation of mathematical principles and their real-world applications.

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6 Here’s what else to consider

This is a space to share examples, stories, or insights that don’t fit into any of the previous sections. What else would you like to add?

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Abhiram Sripat

COO at Florence Quantum Labs | Innovating Climate Solutions with Quantum Machine Learning
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When considering region division in origami, also think about the choice of paper and its properties, such as thickness, texture, and size, as these can affect the precision and ease of folding. Pay attention to the symmetry and alignment of creases to ensure uniformity in the divided sections. Explore different folding techniques, such as accordion folds or pleats, to achieve various effects and patterns. Additionally, consider the end purpose of the design, whether for artistic display, educational tools, or practical applications, as this will guide the complexity and precision required in the region division process.

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How do you solve complex origami problems using geometric reasoning and logic?

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1 Basic folds and axioms

2 Regular polygons

3 Angle bisectors

4 Region division

5 Applications and benefits

6 Here’s what else to consider

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