{"id":2074,"date":"2026-10-11T11:56:53","date_gmt":"2026-10-11T11:56:53","guid":{"rendered":"https:\/\/stadiaorg.com\/?page_id=2074"},"modified":"2026-10-11T11:56:56","modified_gmt":"2026-10-11T11:56:56","slug":"numbers-and-universe","status":"publish","type":"page","link":"https:\/\/stadiaorg.com\/it\/numbers-and-universe\/","title":{"rendered":"Numeri e Universo"},"content":{"rendered":"<style>.smh{--ink:#211b40;--muted:#625c75;--purple:#6650bd;max-width:1260px;margin:38px auto 60px;padding:0 20px;font-family:system-ui,-apple-system,\"Segoe UI\",sans-serif;color:var(--ink);line-height:1.75}.smh,.smh *{box-sizing:border-box}.smh p{margin:0 0 16px}.smh h1,.smh h2,.smh h3{font-family:inherit;line-height:1.18;letter-spacing:-.035em;color:var(--ink);margin:0 0 18px}.smh h2{font-size:clamp(28px,3vw,39px)}.smh h3{font-size:24px}.smh a{color:#5841a8}.smh .smh-kicker{font-size:11px;letter-spacing:.14em;font-weight:800;text-transform:uppercase;margin-bottom:15px}.smh 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title=\"English\">\r\n                            <img decoding=\"async\" src=\"https:\/\/stadiaorg.com\/wp-content\/plugins\/translatepress-multilingual\/assets\/flags\/4x3\/en_US.svg\" class=\"trp-flag-image\" alt=\"\" role=\"presentation\" loading=\"lazy\" \/>                                                            <span class=\"trp-language-item-name\" data-no-translation>English<\/span>\r\n                                                                                <\/a>\r\n                                            <a class=\"trp-language-item\" href=\"https:\/\/stadiaorg.com\/it\/wp-json\/wp\/v2\/pages\/2074#TRPLINKPROCESSED\" title=\"German\">\r\n                            <img decoding=\"async\" src=\"https:\/\/stadiaorg.com\/wp-content\/plugins\/translatepress-multilingual\/assets\/flags\/4x3\/de_DE_formal.svg\" class=\"trp-flag-image\" alt=\"\" role=\"presentation\" loading=\"lazy\" \/>                                                            <span class=\"trp-language-item-name\" data-no-translation>German<\/span>\r\n                                                                                <\/a>\r\n                                            <a class=\"trp-language-item\" href=\"https:\/\/stadiaorg.com\/it\/wp-json\/wp\/v2\/pages\/2074#TRPLINKPROCESSED\" title=\"French\">\r\n                            <img decoding=\"async\" src=\"https:\/\/stadiaorg.com\/wp-content\/plugins\/translatepress-multilingual\/assets\/flags\/4x3\/fr_FR.svg\" class=\"trp-flag-image\" alt=\"\" role=\"presentation\" loading=\"lazy\" \/>                                                            <span class=\"trp-language-item-name\" data-no-translation>French<\/span>\r\n                                                                                <\/a>\r\n                                            <a class=\"trp-language-item\" href=\"https:\/\/stadiaorg.com\/it\/wp-json\/wp\/v2\/pages\/2074#TRPLINKPROCESSED\" title=\"Spanish\">\r\n                            <img decoding=\"async\" src=\"https:\/\/stadiaorg.com\/wp-content\/plugins\/translatepress-multilingual\/assets\/flags\/4x3\/es_ES.svg\" class=\"trp-flag-image\" alt=\"\" role=\"presentation\" loading=\"lazy\" \/>                                                            <span class=\"trp-language-item-name\" data-no-translation>Spanish<\/span>\r\n                                                                                <\/a>\r\n                                                <\/div>\r\n        <\/div>\r\n\r\n    <\/div>\r\n<\/div>\r\n<header class=\"smh-hero\"><div><p class=\"smh-kicker\">STADIA \/ NUMBERS &amp; THE UNIVERSE<\/p><h1 id=\"smh-1title\">The universe.<br><em>Written in equations.<\/em><\/h1><p class=\"smh-motto\">Understand the symbols.<br>Discover the physical relationship.<\/p><p class=\"smh-lead\">From motion and gravity to Einstein\u2019s relativity: learn what a formula means, how to read it and where it applies.<\/p><a class=\"smh-button\" href=\"#smh-1laws\">Explore the equations \u2193<\/a><a class=\"smh-quiet\" href=\"#smh-1method\">How to understand a formula \u2192<\/a><\/div><figure class=\"smh-visual\"><svg viewBox=\"0 0 480 480\" role=\"img\" aria-label=\"Schematic diagram of orbits and a curved grid\"><rect width=\"480\" height=\"480\" rx=\"20\" fill=\"#211847\"\/><g stroke=\"#7965aa\" fill=\"none\" stroke-width=\"1.5\"><path d=\"M40 100Q240 150 440 100 M40 170Q240 290 440 170 M40 240Q240 400 440 240 M40 310Q240 410 440 310 M90 55Q160 240 90 425 M170 55Q220 240 170 425 M250 55Q290 240 250 425 M330 55Q350 240 330 425 M410 55V425\"\/><\/g><g fill=\"none\" stroke=\"#c5b1eb\" stroke-width=\"2\"><ellipse cx=\"240\" cy=\"230\" rx=\"175\" ry=\"80\" transform=\"rotate(-30 240 230)\"\/><ellipse cx=\"240\" cy=\"230\" rx=\"115\" ry=\"140\" transform=\"rotate(35 240 230)\"\/><\/g><circle cx=\"240\" cy=\"230\" r=\"23\" fill=\"#d3ac68\"\/><circle cx=\"371\" cy=\"131\" r=\"8\" fill=\"#e7d8ff\"\/><\/svg><figcaption>A schematic illustration. Spacetime is four-dimensional; this drawing is not a physical map of its curvature.<\/figcaption><\/figure><\/header>\r\n<nav class=\"smh-nav\" aria-label=\"Physics and mathematics\"><a href=\"#smh-1method\">Reading formulas<\/a><a href=\"#smh-1laws\"><span lang=\"it\" translate=\"no\" data-no-translation data-no-dynamic-translation>Sei equazioni<\/span><\/a><a href=\"#smh-1foundations\">Mathematical foundations<\/a><a href=\"#smh-1cosmology\">Cosmology<\/a><\/nav>\r\n<div class=\"smh-intro\"><div><p class=\"smh-kicker\">PHYSICAL LAWS \/ MATHEMATICAL MODELS<\/p><h2>A formula expresses<br>a relationship.<\/h2><\/div><div><p>The symbols in a physical equation describe measurable quantities or mathematical structures. Understanding the equation means connecting the symbols with an experiment, a prediction or a model.<\/p><p>Different theories work in different regimes. Newtonian mechanics describes many everyday situations very well. Relativity becomes essential for high speeds, strong gravity and precise spacetime measurements.<\/p><p class=\"smh-note\">There is no experimentally established single equation that fully describes every phenomenon. Quantum theory and general relativity remain distinct frameworks, and a complete account of quantum gravity is still an open problem.<\/p><\/div><\/div>\r\n<section class=\"smh-method\" id=\"smh-1method\"><p class=\"smh-kicker\">THE METHOD \/ UNDERSTAND BEFORE CALCULATING<\/p><h2>How to read a physical formula<\/h2><ol><li><strong>Name the quantities.<\/strong><span lang=\"it\" translate=\"no\" data-no-translation data-no-dynamic-translation> Che cosa significa ciascun simbolo e che cosa si pu\u00f2 misurare?<\/span><\/li><li><strong>Check the units.<\/strong> Both sides must describe the same kind of quantity.<\/li><li><strong>Read the relationship aloud.<\/strong><span lang=\"it\" translate=\"no\" data-no-translation data-no-dynamic-translation> Quali grandezze si moltiplicano, si dividono o sono elevate al quadrato?<\/span><\/li><li><strong>Change one quantity.<\/strong> Predict the effect while keeping the others fixed.<\/li><li><strong>Check the assumptions.<\/strong> Is the model classical or relativistic? Is the field weak? Is the speed constant?<\/li><\/ol><p class=\"smh-note\">Dimensional consistency is necessary but does not prove that a physical law is correct. Predictions must be compared with observations.<\/p><\/section>\r\n<section class=\"smh-section\" id=\"smh-1laws\"><p class=\"smh-kicker\"><span lang=\"it\" translate=\"no\" data-no-translation data-no-dynamic-translation>DA NEWTON A EINSTEIN<\/span><\/p><h2><span lang=\"it\" translate=\"no\" data-no-translation data-no-dynamic-translation>Sei equazioni, spiegate passo dopo passo.<\/span><\/h2><div class=\"smh-grid\"><article class=\"smh-card\"><span class=\"smh-number\" aria-hidden=\"true\">01<\/span><h3>Motion and change<\/h3><div class=\"smh-equation\" translate=\"no\" data-no-translation data-no-dynamic-translation>F = m a<\/div><details open><summary>The symbols<\/summary><p>F: net force in newtons; m: constant mass in kilograms; a: acceleration in metres per second squared.<\/p><\/details><details open><summary>Read the relationship<\/summary><p>Read it as: the net force equals the mass multiplied by the acceleration. Acceleration describes a change of velocity, not velocity itself.<\/p><\/details><details><summary>A worked example or first reading step<\/summary><p>A net force of 10 N acting on a mass of 2 kg gives an acceleration of 5 m\/s\u00b2. Doubling the force doubles the acceleration if the mass is unchanged.<\/p><\/details><details><summary>Conditions and limits<\/summary><p>For constant mass in an inertial frame, within classical mechanics. Force and acceleration are vectors; this scalar form can describe one chosen direction.<\/p><\/details><\/article><article class=\"smh-card\"><span class=\"smh-number\" aria-hidden=\"true\">02<\/span><h3>Newtonian gravity<\/h3><div class=\"smh-equation\" translate=\"no\" data-no-translation data-no-dynamic-translation>F = G m\u2081 m\u2082 \/ r\u00b2<\/div><details open><summary>The symbols<\/summary><p><span lang=\"it\" translate=\"no\" data-no-translation data-no-dynamic-translation>G: costante di gravitazione universale; m\u2081 e m\u2082: le due masse; r: distanza tra i loro centri, nel caso di corpi sferici.<\/span><\/p><\/details><details open><summary>Read the relationship<\/summary><p>The force grows with each mass and decreases with the square of separation. The formula gives the magnitude of an attractive force.<\/p><\/details><details><summary>A worked example or first reading step<\/summary><p><span lang=\"it\" translate=\"no\" data-no-translation data-no-dynamic-translation>Mantieni costanti entrambe le masse e raddoppia la distanza r. Il denominatore diventa 4r\u00b2, quindi la forza si riduce a un quarto del valore precedente.<\/span><\/p><\/details><details><summary>Conditions and limits<\/summary><p>Exact for ideal point masses in Newtonian gravity; also applies to separated spherically symmetric bodies. It is a useful approximation in weak gravitational fields with slow motion.<\/p><\/details><\/article><article class=\"smh-card\"><span class=\"smh-number\" aria-hidden=\"true\">03<\/span><h3>Special relativity: time<\/h3><div class=\"smh-equation\" translate=\"no\" data-no-translation data-no-dynamic-translation>\u0394t = \u03b3 \u0394\u03c4; \u03b3 = 1 \/ \u221a(1 \u2212 v\u00b2\/c\u00b2)<\/div><details open><summary>The symbols<\/summary><p><span lang=\"it\" translate=\"no\" data-no-translation data-no-dynamic-translation>\u0394\u03c4: tempo proprio misurato da un orologio tra due eventi lungo la sua traiettoria; \u0394t: intervallo di tempo in un sistema di riferimento inerziale; v: velocit\u00e0 costante dell\u2019orologio in quel sistema; c: velocit\u00e0 della luce nel vuoto.<\/span><\/p><\/details><details open><summary>Read the relationship<\/summary><p><span lang=\"it\" translate=\"no\" data-no-translation data-no-dynamic-translation>Un osservatore inerziale assegna un intervallo di tempo maggiore di quello misurato da un orologio che si muove a velocit\u00e0 costante rispetto a lui. L\u2019orologio in movimento misura normalmente il proprio tempo proprio.<\/span><\/p><\/details><details><summary>A worked example or first reading step<\/summary><p><span lang=\"it\" translate=\"no\" data-no-translation data-no-dynamic-translation>Per v = 0,6c, \u03b3 = 1\/\u221a0,64 = 1,25. Se l\u2019orologio in movimento misura 8 secondi tra due eventi, il sistema inerziale assegna un intervallo di 10 secondi.<\/span><\/p><\/details><details><summary>Conditions and limits<\/summary><p>This calculation assumes constant relative speed and neglects gravity. Comparing twins who reunite requires analysing their complete paths, not simply applying this formula symmetrically.<\/p><\/details><\/article><article class=\"smh-card\"><span class=\"smh-number\" aria-hidden=\"true\">04<\/span><h3>Mass and energy<\/h3><div class=\"smh-equation\" translate=\"no\" data-no-translation data-no-dynamic-translation>E\u2080 = m c\u00b2<\/div><details open><summary>The symbols<\/summary><p><span lang=\"it\" translate=\"no\" data-no-translation data-no-dynamic-translation>E\u2080: energia a riposo in joule; m: massa invariante in chilogrammi; c = 299 792 458 m\/s nel vuoto.<\/span><\/p><\/details><details open><summary>Read the relationship<\/summary><p><span lang=\"it\" translate=\"no\" data-no-translation data-no-dynamic-translation>Un sistema possiede un\u2019energia a riposo associata alla sua massa. Questo non significa che tutta quell\u2019energia sia disponibile per essere estratta.<\/span><\/p><\/details><details><summary>A worked example or first reading step<\/summary><p><span lang=\"it\" translate=\"no\" data-no-translation data-no-dynamic-translation>Per m = 1 kg, E\u2080 \u00e8 circa 8,99 \u00d7 10\u00b9\u2076 J. Elevare c al quadrato significa elevare al quadrato anche la sua unit\u00e0 di misura: kg \u00d7 m\u00b2\/s\u00b2 corrisponde a un joule.<\/span><\/p><\/details><details><summary>Conditions and limits<\/summary><p><span lang=\"it\" translate=\"no\" data-no-translation data-no-dynamic-translation>E\u2080 \u00e8 l\u2019energia a riposo. Pi\u00f9 in generale, E\u00b2 = p\u00b2c\u00b2 + m\u00b2c\u2074, dove p \u00e8 il modulo della quantit\u00e0 di moto. Un fotone ha massa a riposo nulla e pu\u00f2 comunque trasportare energia.<\/span><\/p><\/details><\/article><article class=\"smh-card\"><span class=\"smh-number\" aria-hidden=\"true\">05<\/span><h3>General relativity: spacetime<\/h3><div class=\"smh-equation\" translate=\"no\" data-no-translation data-no-dynamic-translation>G\u03bc\u03bd + \u039bg\u03bc\u03bd = (8\u03c0G\/c\u2074)T\u03bc\u03bd<\/div><details open><summary>The symbols<\/summary><p>G\u03bc\u03bd: Einstein tensor, built from spacetime curvature; g\u03bc\u03bd: metric tensor; \u039b: cosmological constant; T\u03bc\u03bd: stress\u2013energy tensor; G without indices: gravitational constant.<\/p><\/details><details open><summary>Read the relationship<\/summary><p>This is a set of tensor equations relating spacetime geometry to energy, momentum and stresses. The indices label spacetime components: the equation is not a single ordinary multiplication exercise.<\/p><\/details><details><summary>A worked example or first reading step<\/summary><p>Begin by separating the two sides: the left describes geometry, while the right describes physical sources. Then study how a metric defines distances and time intervals before attempting a full calculation.<\/p><\/details><details><summary>Conditions and limits<\/summary><p>General relativity is a classical theory. It is not a complete quantum theory of gravity. The equation requires a matter model and initial or boundary conditions; notation and sign conventions must be stated.<\/p><\/details><\/article><article class=\"smh-card\"><span class=\"smh-number\" aria-hidden=\"true\">06<\/span><h3>The expanding universe<\/h3><div class=\"smh-equation\" translate=\"no\" data-no-translation data-no-dynamic-translation>v \u2248 H\u2080 d<\/div><details open><summary>The symbols<\/summary><p>v: recession speed inferred in the nearby-universe approximation; d: distance; H\u2080: present expansion rate.<\/p><\/details><details open><summary>Read the relationship<\/summary><p>For nearby galaxies following the large-scale expansion, recession speed is approximately proportional to distance. H\u2080 is the present value of an expansion rate that evolves with cosmic time.<\/p><\/details><details><summary>A worked example or first reading step<\/summary><p>Using 70 km\/s\/Mpc purely as an illustrative value: at 10 Mpc, the expansion contribution is approximately 700 km\/s. Local peculiar motions can also affect observations.<\/p><\/details><details><summary>Conditions and limits<\/summary><p>This nearby relation is not a universal conversion of any redshift into speed. At large distances, cosmological distances and redshifts require an expansion model. The illustrative H\u2080 value is not a claim of a settled measurement.<\/p><\/details><\/article><\/div><\/section>\r\n<section class=\"smh-method\" id=\"smh-1foundations\"><p class=\"smh-kicker\">THE MATHEMATICS BEHIND THE PHYSICS<\/p><h2>Build the foundations in order.<\/h2><div class=\"smh-method-grid\"><article><h3>Algebra and proportions<\/h3><p>Rearrange an equation and isolate an unknown. In F = ma, dividing both sides by m gives a = F\/m when m is nonzero.<\/p><\/article><article><h3>Vectors and derivatives<\/h3><p>A vector has magnitude and direction. Velocity is the time derivative of position; acceleration is the time derivative of velocity. Derivatives describe local rates of change.<\/p><\/article><article><h3>Geometry and tensors<\/h3><p>A metric describes spacetime intervals. A tensor represents relationships whose components change systematically under a change of coordinates. This is the language used in general relativity.<\/p><\/article><\/div><p class=\"smh-note\">The route to the field equations includes calculus, linear algebra and differential geometry. It is possible to understand their physical role before mastering a full tensor calculation.<\/p><\/section>\r\n<section class=\"smh-section\" id=\"smh-1cosmology\"><p class=\"smh-kicker\">FROM SPACETIME TO COSMOLOGY<\/p><h2>How does the expansion evolve?<\/h2><p>The cosmological scale factor a(t) describes how separations between ideal comoving locations change. The expansion rate is H(t) = a\u0307(t)\/a(t), where the dot means a derivative with respect to cosmic time.<\/p><div class=\"smh-equation\" translate=\"no\" data-no-translation data-no-dynamic-translation>H\u00b2 = (8\u03c0G\/3)\u03c1 + \u039bc\u00b2\/3<\/div><p>This is the first Friedmann equation for a spatially flat, homogeneous and isotropic model. Here \u03c1 is the mass-equivalent density of matter and radiation; the cosmological-constant contribution is written separately to avoid counting it twice.<\/p><p>It relates the expansion rate to the contents of the model. Other equations describe how those contents evolve; one equation alone does not determine the complete history of the universe.<\/p><p class=\"smh-note\">Spatial flatness is an assumption in this displayed form, not a statement that spacetime has no curvature. A more general Friedmann equation includes a spatial-curvature term.<\/p><\/section>\r\n<section class=\"smh-laboratory\"><div><p class=\"smh-kicker\">ANOTHER WAY TO PRACTISE MATHEMATICS<\/p><h2>Patterns, repetition<br>and fractals.<\/h2><p>Our existing laboratory explores mathematical patterns. It complements this physics introduction by letting you practise prediction, calculation and explanation.<\/p><a class=\"smh-button\" href=\"https:\/\/stadiaorg.com\/it\/stadia-numbers-the-universe\/\">Open the patterns laboratory \u2192<\/a><\/div><div class=\"smh-lab-list\"><div><span>01<\/span><strong>Predict<\/strong><p>Change one quantity and predict the result before calculating.<\/p><\/div><div><span>02<\/span><strong>Calculate<\/strong><p>Keep track of the operations, the symbols and their meanings.<\/p><\/div><div><span>03<\/span><strong>Explain<\/strong><p>Connect the numerical answer with the underlying relationship.<\/p><\/div><\/div><\/section>\r\n<footer class=\"smh-footer\"><h3>Further study<\/h3><p><a href=\"https:\/\/www.einstein-online.info\/en\/spotlights\/\" target=\"_blank\" rel=\"noopener noreferrer\">Relativity explanations \u00b7 Einstein Online, Albert Einstein Institute \u2197<\/a><\/p><p><a href=\"https:\/\/science.nasa.gov\/mission\/hubble\/science\/science-behind-the-discoveries\/hubble-cosmological-redshift\/\" target=\"_blank\" rel=\"noopener noreferrer\">Cosmological redshift and expansion \u00b7 NASA \u2197<\/a><\/p><p>Equations here introduce selected physical theories. Simplifying assumptions are stated with each example. Source websites retain their own languages.<\/p><\/footer>\r\n<\/section>\r\n\n","protected":false},"excerpt":{"rendered":"","protected":false},"author":44,"featured_media":0,"parent":0,"menu_order":0,"comment_status":"closed","ping_status":"closed","template":"","meta":{"footnotes":""},"class_list":["post-2074","page","type-page","status-publish","hentry"],"_links":{"self":[{"href":"https:\/\/stadiaorg.com\/it\/wp-json\/wp\/v2\/pages\/2074","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/stadiaorg.com\/it\/wp-json\/wp\/v2\/pages"}],"about":[{"href":"https:\/\/stadiaorg.com\/it\/wp-json\/wp\/v2\/types\/page"}],"author":[{"embeddable":true,"href":"https:\/\/stadiaorg.com\/it\/wp-json\/wp\/v2\/users\/44"}],"replies":[{"embeddable":true,"href":"https:\/\/stadiaorg.com\/it\/wp-json\/wp\/v2\/comments?post=2074"}],"version-history":[{"count":1,"href":"https:\/\/stadiaorg.com\/it\/wp-json\/wp\/v2\/pages\/2074\/revisions"}],"predecessor-version":[{"id":2075,"href":"https:\/\/stadiaorg.com\/it\/wp-json\/wp\/v2\/pages\/2074\/revisions\/2075"}],"wp:attachment":[{"href":"https:\/\/stadiaorg.com\/it\/wp-json\/wp\/v2\/media?parent=2074"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}