Prayer of St. Pio of Pietrelcina after Holy Communion
Stay with me, Lord, for it is necessary to have You present so that I do not forget You. You know how easily I abandon You.
Stay with me, Lord, because I am weak and I need Your strength, that I may not fall so often.
Stay with me, Lord, for You are my life, and without You, I am without fervor.
Stay with me, Lord, for You are my light, and without You, I am in darkness.
Stay with me, Lord, to show me Your will.
Stay with me, Lord, so that I hear Your voice and follow You.
Stay with me, Lord, for I desire to love You very much, and always be in Your company.
Stay with me, Lord, if You wish me to be faithful to You.
Stay with me, Lord, for as poor as my soul is, I want it to be a place of consolation for You, a nest of love.
Stay with me, Jesus, for it is getting late and the day is coming to a close, and life passes; death, judgment, eternity approaches. It is necessary to renew my strength, so that I will not stop along the way and for that, I need You. It is getting late and death approaches, I fear the darkness, the temptations, the dryness, the cross, the sorrows. O how I need You, my Jesus, in this night of exile!
Stay with me tonight, Jesus, in life with all its dangers. I need You.
Let me recognize You as Your disciples did at the breaking of the bread, so that the Eucharistic Communion be the Light which disperses the darkness, the force which sustains me, the unique joy of my heart.
Stay with me, Lord, because at the hour of my death, I want to remain united to You, if not by communion, at least by grace and love.
Stay with me, Jesus, I do not ask for divine consolation, because I do not merit it, but the gift of Your Presence, oh yes, I ask this of You!
Stay with me, Lord, for it is You alone I look for, Your Love, Your Grace, Your Will, Your Heart, Your Spirit, because I love You and ask no other reward but to love You more and more.
With a firm love, I will love You with all my heart while on earth and continue to love You perfectly during all eternity. Amen.
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That is that one can be cut into finitely many polygon pieces and be rearranged to obtain the second polygon.
In geometry, the Wallace–Bolyai–Gerwien theorem, named after William Wallace, Farkas Bolyai and Paul Gerwien, states that any two simple polygons of equal area are equidecomposable; i.e. one can cut the first into finitely many polygonal pieces and rearrange the pieces to obtain the second polygon.
"Rearrangement" means that one may apply a translation and a rotation to every polygonal piece.
Unlike the generalized solution to Tarski's circle-squaring problem, the axiom of choice is not required for the proof, and the decomposition and reassembly can actually be carried out "physically"; the pieces can, in theory, be cut with scissors from paper and reassembled by hand.
The theorem can be understood in two steps. First, every polygon can be cut into triangles: for convex polygons this is immediate, by cutting off each vertex in turn, while for concave polygons this requires more care. Each of these triangles can then be converted to a right triangle, by dropping an altitude (that is, drawing a line perpendicular to the triangle's base and through the top vertex). This is sufficient to easily compute the area, as each right triangle is half a rectangle, or alternatively can be cut halfway up to be reassembled into a rectangle. The second and subtler step is that each right triangle (or equivalently rectangle) can be decomposed into a rectangle with a side of a given (unit) length. Once this is proven, it follows that every polygon can be decomposed into a rectangle with unit width and height equal to its area, which proves the theorem._
#maths #mathematics #geometry #amazing #science
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