# Learn Origami Mathematics and Geometry with The Greatest Dream Origami Toshi

## The Greatest Dream Origami Toshi

Origami is the art of folding paper into various shapes and forms, such as animals, flowers, stars, and more. Origami is popular around the world for its beauty, simplicity, creativity, and fun. But origami is more than just a hobby or a craft. It is also a science, a mathematics, a philosophy, and a dream. And no one knows this better than Toshikazu Kawasaki, the greatest dream origami toshi.

## The Greatest Dream Origami Toshi

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Toshikazu Kawasaki is a Japanese mathematician and origamist who has created hundreds of original origami models, including the famous Kawasaki rose and the Kawasaki snowflake. He is also known for his dream of creating a perfect origami model that can be folded from any shape of paper, with any number of sides, without any cuts or glue. In this article, we will explore the fascinating world of origami and Toshikazu Kawasaki, and learn how to make some of his amazing origami models.

## What is origami and why is it popular?

Origami is derived from the Japanese words "ori" meaning folding and "kami" meaning paper. It is believed that origami originated in China around the first or second century AD, when paper was invented. Paper was then brought to Japan by Buddhist monks in the sixth century AD, where it was used for religious ceremonies and rituals. Origami became a popular art form in Japan during the Edo period (1603-1868), when paper was more widely available and affordable.

Origami is popular for many reasons. Some people enjoy origami as a relaxing and meditative activity that helps them focus and calm their minds. Some people like origami as a creative and expressive outlet that allows them to make beautiful and unique artworks. Some people appreciate origami as a scientific and mathematical challenge that involves geometry, symmetry, angles, fractions, and patterns. Some people love origami as a cultural and educational tool that teaches them about different countries, animals, plants, legends, and stories.

### The history and evolution of origami

Origami has a long and rich history that spans across different cultures and eras. Some of the earliest examples of origami are found in ancient China, where paper was used to make ceremonial objects such as boats, lanterns, boxes, and envelopes. In Japan, origami was used to make sacred offerings such as cranes, frogs, butterflies, and flowers. In Europe, origami was introduced by travelers and traders who brought paper from Asia. Origami was also influenced by other paper arts such as kirigami (cutting paper), quilling (rolling paper), and papier-mâché (molding paper).

Origami has evolved over time as new techniques, styles, and materials were developed. Some of the most influential origami masters include Akira Yoshizawa, who pioneered the use of wet-folding and created thousands of realistic origami models; Robert J. Lang, who applied computer algorithms and engineering principles to design complex and realistic origami models; and Toshikazu Kawasaki, who discovered the mathematical formulas and methods for folding regular polygons from paper.

### The benefits and challenges of origami

Origami has many benefits for people of all ages and backgrounds. Origami can improve one's cognitive skills, such as spatial reasoning, logic, problem-solving, memory, and concentration. Origami can also enhance one's artistic skills, such as creativity, imagination, color sense, and aesthetics. Origami can also foster one's emotional skills, such as patience, perseverance, confidence, and satisfaction. Origami can also promote one's social skills, such as communication, cooperation, and friendship.

Origami also has some challenges that require one's attention and effort. Origami can be frustrating and difficult at times, especially when the paper is too thick or too thin, when the folds are not precise or symmetrical, when the model is too complex or too simple, or when the instructions are not clear or complete. Origami can also be expensive and wasteful at times, especially when the paper is rare or imported, when the model is large or intricate, or when the paper is damaged or discarded.

## Who is Toshikazu Kawasaki and what is his dream?

Toshikazu Kawasaki is a Japanese mathematician and origamist who was born in 1955 in Sapporo, Hokkaido. He graduated from Hokkaido University with a degree in mathematics in 1979. He then worked as a high school teacher for 10 years before becoming a lecturer at Sasebo College of Technology in 1989. He is currently a professor at Kawasaki University of Medical Welfare in Kurashiki.

Toshikazu Kawasaki is best known for his contributions to the field of origami mathematics and geometry. He has published several books and papers on origami theory and practice, such as "The Science of Origami", "Origami for the Connoisseur", "Origami Dream World", and "The Greatest Dream Origami". He has also created hundreds of original origami models that are admired for their elegance, beauty, and simplicity.

Toshikazu Kawasaki has a dream of creating a perfect origami model that can be folded from any shape of paper, with any number of sides, without any cuts or glue. He calls this model the "universal molecule". He believes that this model would be the ultimate expression of origami as an art and a science. He has been working on this dream for over 30 years, and he has not given up yet.

### The biography and achievements of Toshikazu Kawasaki

Toshikazu Kawasaki was fascinated by origami since he was a child. He learned how to fold origami from his mother and his grandmother, who taught him traditional Japanese origami models such as cranes, boats, boxes, and flowers. He also learned how to fold origami from books and magazines that he borrowed from libraries and bookstores. He started to create his own origami models when he was in elementary school.

Toshikazu Kawasaki became interested in mathematics when he was in high school. He enjoyed studying geometry, algebra, calculus, and trigonometry. He also liked to solve puzzles and problems that involved numbers, shapes, patterns, and logic. He decided to pursue mathematics as his major in college.

Toshikazu Kawasaki combined his passion for origami and mathematics when he was in college. He began to explore the mathematical principles and formulas that govern the folding of paper into various shapes and forms. He also began to experiment with different types of paper and folding techniques to create new and original origami models.

Toshikazu Kawasaki made several breakthroughs in origami mathematics and geometry that earned him recognition and respect from the origami community. Some of his most notable achievements include:

Discovering the Kawasaki theorem in 1989, which states that the sum of the odd angles around a vertex on a flat-folded crease pattern is equal to 180 degrees.

Discovering the Kawasaki-Justin theorem in 1997, which states that a flat-foldable crease pattern with only one vertex must have an even number of creases.

Discovering the Kawasaki formula in 2003, which gives the exact angle for folding any regular polygon from a square paper.

Discovering the Kawasaki method in 2005, which gives a general procedure for folding any regular polygon from any shape of paper.

Twist the center of the star with your fingers or a tool to create a spiral. Twist it firmly and tightly, making sure it stays in place.

Fold and tuck some edges and corners of the paper to create details such as thorns, leaves, stems, or buds. You can use scissors to cut some parts of the paper if you want.

Your Kawasaki rose is complete! You can use it as a decoration, a gift, or a part of a larger origami structure.

### How to make a Kawasaki snowflake

The Kawasaki snowflake is another famous origami model by Toshikazu Kawasaki. It was invented by Toshikazu Kawasaki in 2005, and it is based on his mathematical method for folding any regular polygon from any shape of paper. The Kawasaki snowflake has a symmetrical appearance with six arms and six pockets. It is also modular, meaning that it can be combined with other snowflakes or other origami models to create complex structures.

#### Step 1: Fold a hexagon from a square paper

The first step to make a Kawasaki snowflake is to fold a hexagon from a square paper. This is based on the Kawasaki method, which gives a general procedure for folding any regular polygon from any shape of paper. The method involves finding the center of the paper, dividing the paper into equal parts, and folding along bisectors of angles. Here are the instructions:

If you are working with colored paper, start with the color you want the finished snowflake to be facing up. (if you are working with printer paper, you may have a diagonal crease as a result of creating a perfect square, this is fine) Fold the paper in half from the edge, then turn the paper 90 degrees and repeat.

Unfold the paper and mark the center point of the paper by pinching it lightly with your fingers.

Fold one corner of the paper to the center point. Do this on all four corners.

Unfold the paper and mark the midpoint of each edge by pinching it lightly with your fingers.

Fold one edge of the paper to the midpoint of the opposite edge. Do this on all four edges.

Unfold the paper and mark the intersection point of each pair of creases by pinching it lightly with your fingers.

Fold one corner of the paper to the intersection point that is closest to it. Do this on all four corners.

Unfold the paper and mark the intersection point of each pair of creases by pinching it lightly with your fingers.

Fold one edge of the paper to the intersection point that is closest to it. Do this on all four edges.

Unfold the paper and mark the center point of the paper by pinching it lightly with your fingers.

Draw a line from the center point to one of the intersection points that is closest to it. This line will be one side of the hexagon.

Use a protractor to measure an angle of 120 degrees from the line you just drew. Draw another line from the center point along this angle. This line will be another side of the hexagon.

Repeat step 9 until you have drawn six lines from the center point, forming a hexagon.

Cut out the hexagon along the lines you drew. You can also fold and tear the paper along the lines if you prefer.

#### Step 2: Fold the six arms of the snowflake

The second step to make a Kawasaki snowflake is to fold the six arms of the snowflake. This is based on Kawasaki's modular technique, which creates units that can be joined together to form a larger structure. The arms of the snowflake are folded from six identical units, each made from one sixth of the hexagon. Here are the instructions:

Fold the hexagon in half along one of its diameters, then unfold.

Fold one edge of the hexagon to the crease you just made, then unfold.

Repeat step 2 on all six edges of the hexagon.

Cut or tear along each crease you just made, dividing the hexagon into six equal parts. Each part will be a trapezoid with two right angles and two acute angles.

Take one trapezoid and fold it in half along its longer base, then unfold.

Fold one edge of the trapezoid to the crease you just made, then unfold.

Repeat step 6 on all four edges of the trapezoid.

Make a reverse fold along each crease you just made, tucking each edge inside.

Your model should now look like a parallelogram with two pockets on each side.

Fold one corner of the parallelogram to the opposite corner, then unfold.

Make a squash fold along each crease you just made, opening up each pocket and flattening it down along a new crease.

Your model should now look like a kite with two flaps on each side.

Fold one flap of the kite to the opposite flap, then unfold.

Make a petal fold along each crease you just made, lifting each flap up and flattening it down along two diagonal creases and one horizontal crease.

Your model should now look like a diamond with two points and two pockets on each side.

Repeat steps 5-15 on all five remaining trapezoids. You will have six identical units that will form the arms of the snowflake.

#### Step 3: Unfold and flatten the snowflake

The third step to make a Kawasaki snowflake is to unfold and flatten the snowflake. This is based on Kawasaki's flat-folding technique, which creates models that can be collapsed into a single plane. The snowflake is unfolded by reversing some of the folds and flattening some of the pockets. The snowflake is flattened by aligning some of the folds and creases. Here are the instructions:

Take two units and insert one point of one unit into one pocket of the other unit. Make sure the points and pockets are aligned.

Repeat step 1 until you have connected all six units into a ring. The ring should look like a hexagon with six points and six pockets on each side.

Unfold each point of the ring by reversing the petal folds you made earlier. You should see six triangles around a smaller hexagon on each side.

Unfold each pocket of the ring by reversing the squash folds you made earlier. You should see six squares around a smaller hexagon on each side.

Flatten each triangle by aligning its base with the edge of the smaller hexagon. You should see six diamonds around a smaller hexagon on each side.

Flatten each square by aligning its diagonal with the edge of the smaller hexagon. You should see six kites around a smaller hexagon on each side.

Your Kawasaki snowflake is complete! You can use it as a decoration, a gift, or a part of a larger origami structure.

## Conclusion and FAQs

In this article, we have learned about origami and Toshikazu Kawasaki, and how to make some of his amazing origami models. Origami is a fascinating art form that combines beauty, creativity, logic, and mathematics. Toshikazu Kawasaki is a brilliant origamist who has contributed to the field of origami mathematics and geometry, and who has a dream of creating a perfect origami model. We have also learned how to make a Kawasaki rose and a Kawasaki snowflake, two of his most famous origami models that are based on his mathematical formulas and methods.

We hope you have enjoyed this article and have fun folding origami. If you want to learn more about origami and Toshikazu Kawasaki, you can check out his books and papers, or visit his website . You can also find more origami instructions and tutorials on our website . Here are some frequently asked questions about origami and Toshikazu Kawasaki:

### FAQs

Q: What is the difference between origami and kirigami?A: Origami is the art of folding paper without cutting or gluing it. Kirigami is the art of cutting and folding paper, usually with scissors or knives.

Q: What is the best paper for origami?A: There is no definitive answer to this question, as different types of paper may suit different types of origami models. However, some general factors to consider are the size, thickness, color, texture, and quality of the paper. For beginners, it is recommended to use origami paper, which is thin, square, colorful, and easy to fold. For advanced folders, it may be preferable to use other types of paper, such as foil paper, tissue paper, washi paper, or handmade paper.

Q: Who are some other famous origami masters?A: There are many famous origami masters in the world, each with their own style and specialty. Some of them are Akira Yoshizawa (the father of modern origami), Robert J. Lang (a pioneer of origami engineering), Satoshi Kamiya (a master of complex and realistic origami), Tomoko Fuse (a master of modular and geometric origami), Eric Joisel (a master of wet-folding and sculptural origami), John Montroll (a master of animal and mathematical origami), Peter Engel (a master of artistic and expressive origami), Nick Robinson (a master of minimalist and elegant origami), Michael LaFosse (a master of floral and naturalistic origami), and many more.

Q: How can I improve my origami skills?A: The best way to improve your origami skills is to practice regularly and challenge yourself with new models and techniques. You can also learn from books, videos, diagrams, or other folders. You can also join an origami club or community online or offline, where you can share your work, get feedback, ask questions, and learn from others.

Q: How can I use origami in my daily life?A: Origami can be used in many ways in your daily life, such as for decoration, entertainment, education, or therapy. You can use origami to decorate your home, office, or classroom with beautiful and unique models. You can use origami to entertain yourself, your friends, or your family with fun and interactive models. You can use origami to teach or learn about geometry, fractions, symmetry, patterns, or culture. You can use origami to relax, meditate, focus, or express yourself.

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