The mathematical sciences particularly exhibit order symmetry and limitations; and these are the greatest forms of the beautiful.

'Parallelism' can be pointed out in the different parts of a single object, looked at alone; it is even more obvious when one puts several objects of the same kind next to each other. Now if we compare our own lives and customs with these appearances in nature, we shall be astonished to find the same principle repeated.. .When an important event is being celebrated, the people face and move in the same direction. These are parallels following each other..

In many cases a dull proof can be supplemented by a geometric analogue so simple and beautiful that the truth of a theorem is almost seen at a glance.

Narrative: telling stories about the topic and the people involved with it (e.g., the story of Charles Darwin for evolution or of Anne Frank for the Holocaust) 2. Quantitative: using examples connected to the topic (e.g., the puzzle of different numbers and varieties of finches spread across a dozen islands in the Galapagos) 3. Logic: identifying the key elements or units and exploring their logical connections (e.g., how Malthus’s argument about human survival in the face of insufficient resources can be applied to competition among biological species) 4. Existential: addressing big questions, such as the nature of truth or beauty, life and death 5. Aesthetic: examining instances in terms of their artistic properties or capturing the examples themselves in works of art (e.g., observing the diverse shapes of the beaks of finches; analyzing the expressive elements in the trio) 6. Hands-on: working directly with tangible examples (e.g., performing the Figaro trio, breeding fruit flies to observe how traits change over the generations) 7. Cooperative or social: engaging in projects with others where each makes a distinctive contribution to successful execution

It’s really beautiful to observe, as you progress in your mathematical maturity, how everything is somehow connected.

The beauty of mathematics only shows itself to more patient followers.

The history of science is rich in the example of the fruitfulness of bringing two sets of techniques, two sets of ideas, developed in separate contexts for the pursuit of new truth, into touch with one another.

Every true mathematician sees mathematics everywhere—in a child's swing or a pendulum, in the outline shape of a tree and that of its leaves, in the clouds

For many, maths is fundamentally beautiful; indeed many mathematicians won’t be entirely satisfied with their work until it has an elegance, simplicity and grace. Others are drawn by its ‘unreasonable effectiveness’ - its power to explain the world we live in.

The mathematician’s patterns, like the painter’s or the poet’s must be beautiful; the ideas like the colours or the words, must fit together in a harmonious way. Beauty is the first test: there is no permanent place in the world for ugly mathematics.

Those who assert that the mathematical sciences say nothing of the beautiful or the good are in error. For these sciences say and prove a great deal about them; if they do not expressly mention them, but prove attributes which are their results or definitions, it is not true that they tell us nothing about them. The chief forms of beauty are order and symmetry and definiteness, which the mathematical sciences demonstrate in a special degree.

What attracted me so strongly and exclusively to mathematics, apart from the actual content, was particularly the specific nature of the mental processes by which mathematical concepts are handled. This way of deducing and discovering new truths from old ones, and the extraordinary clarity and self-evidence of the theorems, the ingeniousness of the ideas... had an irresistible fascination for me. Beginning from the individual theorems, I grew accustomed to delve more deeply into their relationships and to grasp whole theories as a single entity. That is how I conceived the idea of mathematical beauty...

Mathematics rightly viewed possesses not only truth but supreme beauty.

The beauty of mathematics often makes the subject matter much more attractive and easier to master.