This is a summary of a compilation made from the works of the Mother and Sri Aurobindo. The summary was prepared by a student as part of her learning journey which includes making of the compilation.
What is Beauty?
The physical world is the world of form, and the perfection of form is beauty. Its role is to put the manifested nature in contact with the Eternal through the perfection of the form. 
The Divine expresses itself as knowledge on the mental plane, power on the vital plane, love on the psychic plane, and beauty on the physical plane. When we rise high above, we discover that these four aspects unite with each other in a single consciousness, full of love, luminous, powerful, beautiful, containing all, pervading all. 
The master and mover of our works is the One, the Universal and Supreme, the Eternal and Infinite, the unexpressed and unmanifested Ineffable above us, but He is also the self of all beings, the Master of all worlds, the Light and Guide, he is the All-Beautiful and All-Blissful, the Beloved and the Lover. Though we know it not, we ourselves, are the being of his being, force of his force, conscious with a consciousness derived from his; even our mortal existence is made out of his substance and there is an immortal within us that is a spark of Light and Bliss that are forever.  It is this spark in us, the soul, which turns towards Truth, Good, and Beauty because it is by these things that it grows in stature.  One feels a deep detachment for all things in the world, an imperious need to find something truly beautiful, good, and fresh, this quite naturally develops into a spiritual aspiration.  When the soul becomes visible in the physical, it gives this dignity, this beauty, this majesty; then, even things that have no particular beauty put on a sense of eternal beauty.  Beauty is a thing that moves and evokes a great respect always.  But beauty alone is not sufficient, it wants to become divine. A pure sense of beauty can be acquired only through purification and gets its full power only when surrendered to the divine.   This sense of love, joy, and beauty are also what comprises the very stuff and nature of delight.  To bring the Divine Love and Beauty and Ananda in the world is, indeed, the whole crown and essence of the Yoga. 
There is a beauty of thought, of feeling, this is something we perceive very often. When someone does a noble, unselfish, generous deed, quite spontaneously we say, “It is beautiful” because it gives us a sense of beauty.  True art means the expression of beauty in the material world.  It is a living harmony and beauty that must be expressed in all movements of existence, one with experience of life as a whole.  Beauty, the aesthetic instinct of man, is very important in human education and growth.  Aesthetic conscience is the consciousness of beauty, there are people who move around in life and see landscapes, see people and things and have absolutely no sense of whether it is beautiful or not, it makes no difference at all to them.  An artist should be capable of entering into communion with the Divine and of receiving inspiration about what form or forms ought to be used to express the divine beauty in matter, then art can become a means of realization of beauty, and at the same time a teacher of what beauty ought to be. This is the raison d'être of art. 
Seeking for beauty is instinctive, it’s a kind of harmony which you experience much more than you think, and the true suprarational relation with beauty is not at all a reasonable relation, it completely overpasses reason, it is in contact with a higher realm. But when this instinct is mixed with movements of ignorance and a lack of culture and refinement, one can experience aesthetic pleasure in something which is not beautiful, but which gives one some sort of pleasure, because it is not pure and mixed with all kinds of crude and untrained sensations. Here, reason has some role to play, to educate the taste, to explain why a thing is beautiful. This is the rational stage of beauty, but it is not final, it can very well make mistakes, only it is a little higher in judgment than that of a completely infrarational being who has no reason or understanding of things that it likes, but to realize true beauty one must go beyond.   Everything in the manifested universe is relative, i.e., there is no beauty which may not appear ugly in comparison with a greater beauty, neither no ugliness which isn’t more beautiful than some other ugliness.  But the supramental beauty is something much higher and more perfect, untainted by any ugliness. When the supramental force descends into Matter, this beauty will express itself quite naturally and spontaneously in all forms. 
Why is Beauty Important?
The joy and happiness and satisfaction of beauty that comes from the perception of the Divine everywhere plunges the nature inward towards its meeting with the immanent Divine in the heart’s secret center, felt as an unconditional calling. 
As one offers a flower, a prayer, an act to the Divine, one can offer too a created form of beauty, a song, a poem, an image, a strain of music, and gain through it a contact, a response or an experience.  These things bring us joy but we must go farther on beyond the search of pleasure and personal welfare, not through fear, but through the development of a new sense of beauty, a thirst for truth and light, by understanding that only by widening and illumining ourselves, setting ourselves ablaze with the ardour for progress, an integral peace and enduring happiness can be found. 
For the perfection of the aesthetic mind, it needs to detach from all its crude pleasures and outward conventional aesthetics to discover a self-existent self and spirit of pure and infinite Beauty and Delight, having a light and joy of their own. 
The vital is a strange creature, a being of passion, enthusiasm, and naturally of desire but it is capable of getting enthusiastic over something beautiful and getting converted, it can get carried away, seized by an ambition to do the same thing, and becomes a powerful instrument.  
How to Cultivate Beauty?
For something beautiful to express itself, needs a good instrument - a body with mental, vital, psychic, and physical capacities.  If what you do is beautiful, good, and harmonious, your atmosphere will be beautiful.  When the surroundings, circumstances, atmosphere, and above all the inner attitude are altogether of a low kind, love is reluctant to come and generally does not stay long.  To have a beautiful and harmonious life calls for an imperious need to be sincere and absolute in one’s consecration to the Divine.  Let beauty be your constant ideal, beauty of the soul, of sentiments, of thoughts, of the action, of the work, so that nothing comes out which is not an expression of pure and harmonious beauty.  From beauty to greater beauty, from joy to intenser joy, by an especial adjustment of the senses, is the normal course of gradual divine manifestation in Matter. 
By encouraging the beautiful movements and instincts in children, and becoming sensitive of when there is a beautiful movement, to capture it, cultivate it, and not forget, by holding it constantly and seeking it.   By becoming aware of the ugliness of the movements of the lower nature in us under the pretext that they exist, not to accept and enjoy them, but to get rid of them and create a life of spiritual beauty and perfection.  But even while the lower movements are active, the higher can descend, if there is an opening at all, before the lower movements have definitely gone. 
When the spirit in us is free, then what was behind this soul-force comes out in all its light, beauty, and greatness.  There is, behind all things, a divine beauty, a divine harmony: it is this we must come in contact with, it is this we must express.  This can happen only if one is not stopped by the appearance, physical or moral or aesthetic, but gets behind and comes in touch with the divine Spirit, the Divine Soul in things.  And to have the sense of true beauty which is quite independent of all tastes that belong to a particular race or group - one must have a universal consciousness. Otherwise, there are preferences, if not active ones, there are subconscious preferences, instincts, which is why the widening of consciousness is indispensable. And once you’ve known beauty beyond every form, you can recognize it in any form whatsoever, indifferently.  
The supreme Ananda comes from supreme fulfillment in the supramental manifestation but there is no real reason why there should not be the love and Ananda and beauty on the way also. The secret is in the heart, not the mind—the heart has to open its inner doors and through it, the radiance of the soul looks out in a blaze of trust and self-giving. Before that inner fire, the debates of the mind and its difficulties wither away and the path however long or arduous becomes a sunlit road not only towards but through love and Ananda.