Donald Duck, holding a hunting rifle, passes through a doorway to find that he has entered Mathmagic Land. This fantasy land contains trees with square roots, a stream flowing with numbers, and a walking pencil that plays tic-tac-toe. Interestingly, a geometric bird recites (almost perfectly) the first 15 digits of pi. Donald soon hears the voice of the "True Spirit of Adventure" who will guide Donald on his journey through "the wonderland of mathematics".
Pythagoras and music
Donald is initially not interested in Mathmagic Land, believing that math is for "eggheads". When "Mr. Spirit" suggests a connection between math and music, though, Donald is intrigued. First, Donald discovers the relationships between octaves and string length. Next, Donald finds himself in ancient Greece, where Pythagoras and his contemporaries are discovering these same relationships. Pythagoras (on the harp), a flute player, and a double bass player hold a "jam session" which Donald joins after a few moments using a vase as a bongo drum. Pythagoras' music is, as the Spirit explains, the basis of today's music, and that music would not exist without "eggheads".
The pentagram, the golden section, and the golden rectangle
After shaking hands with Pythagoras, Donald finds on his hand a pentagram, the symbol of the secret Pythagorean society. The Spirit then shows Donald how the mysterious golden section appears in the pentagram. Next, the pentagram is shown to contain the pattern for constructing golden rectangles many times over. According to the Spirit, the golden rectangle has influenced both ancient and modern cultures in many ways.
Architecture and art
Donald learns how the golden rectangle appears in many ancient buildings, such as the Parthenon and the Notre Dame cathedral. Paintings such as the Mona Lisa and various sculptures contain several golden rectangles. The use of the golden rectangle is found in modern architecture, such as the United Nations building in New York City.
The human body and nature
The Spirit shows Donald how the golden rectangle and pentagram are related to the human body and nature, respectively. The human body contains the "ideal proportions" of the golden section; Donald, overinterpreting the Spirit's advice, tries to make his own body fit such a proportion, but his efforts are to no avail; he only manages to fit himself into a pentagon. The pentagram and pentagon are then shown to be found in many flowers and animals, such as the petunia, the star jasmine, the starfish, the wax flower or hoya, and seashells.
Donald learns that mathematics applies not only to nature, architecture, and music, but also to games, including chess, baseball, football, basketball, hopscotch, and three-cushion billiards. Donald even volunteers the game Tiddlywinks, but the Spirit does not pursue this option. Themes of Lewis Carroll's Through the Looking-Glass are scattered throughout the chess scene; Carroll himself was both a writer and a mathematician. The extended billiards scene, which features a non-speaking live actor, describes the calculations involved in the game's "diamond system," and Donald finally learns how to do the calculations but in an exorbitant way with hitting many more cushions than the usual three.
The Spirit then asks Donald to play a mental game, but he finds Donald's mind to be too cluttered with "Antiquated Ideas", "Bungling", "False Concepts", "Superstitions", and "Confusion". After some mental house-cleaning, Donald plays with a circle and a triangle in his mind, and he discovers useful inventions such as the wheel, train, magnifying glass, drill, propeller, and telescope.
Infinity and the future
Donald discovers that pentagrams can be drawn inside each other indefinitely. Therefore, mathematics provides an avenue to consider the infinite. The Spirit states that scientific knowledge and technological advances are unlimited, and the key to unlocking the doors of the future is mathematics. By the end of the film, Donald understands and appreciates the value of mathematics. The film closes with a quote from Galileo: "Mathematics is the alphabet with which God has written the universe".
- Walt Disney's Wonderful World of Color: An Adventure in Color/Mathmagicland
- Donald's Quack Attack, episode #47.
- Mickey's Mouse Tracks, episode #68.
- Walt Disney Treasures: The Chronological Donald, Volume Four
- Donald Duck in Mathmagic Land
- Walt Disney's Fables: Volume 3
- The pencil-bird Donald meets at the start of the movie debuted in Alice in Wonderland: Donald at one point plays the role of Alice himself.