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Gaál Sándor

@gmac4247

Live to win

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basic-geometry.pages.dev
Joined July 2020
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I have to set the record straight regarding an important detail. It doesn't fit into a single post, so it will be a short thread. It's a recurring issue, that some claim that I claim "pi=3.2". That is false. It's the same with the Indiana Pi Bill. ...
Gaál Sándor (@gmac4247) profile pictureGaál Sándor@gmac4247Sep 5
3.2radius² is the exact area of a circle. From that 3.2diameter is the exact circumference. Confirmed via measurements. The wise insist that can't be, because their theoretical circumscribed polygon limits it to about 3.14. Their polygon is not as circumscribed as they think.
3.2radius² is the exact area of a circle.

ALT 3.2radius² is the exact area of a circle.

The circumference of a coin with a diameter of 29 mm measured about 93 mm on a ruler with a thin paper strip.

ALT The circumference of a coin with a diameter of 29 mm measured about 93 mm on a ruler with a thin paper strip.

Arc and tangent line

ALT Arc and tangent line

Arc and tangent line spacing

ALT Arc and tangent line spacing

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I got the focus messed up when I wrote that. The issue is that the Hungarian — my native language — word for the tangent line already suggests single-point tangency. It's that bad. The English word tangent is fine. The concept of single-point tangency is fundamentally wrong.
Basic Geometry (@BasicGeometry) profile pictureBasic Geometry@BasicGeometryJul 25
Replying to @gmac4247
Mainstream geometry insists that a tangent line can always be drawn. A tangent line coincides with a circle only at a single point. Yet, when attempting to circumscribe a 24-sided polygon about a circle, each side coincides with the arc along a significant section.
Attempting to circumscribe a 24-sided polygon about a circle.

ALT Attempting to circumscribe a 24-sided polygon about a circle.

Zooming in on one side of an attempted 24-sided circumscribed polygon about a circle.

ALT Zooming in on one side of an attempted 24-sided circumscribed polygon about a circle.

When attempting to circumscribe a 24-sided polygon about a circle, each side coincides with the arc along a significant section

ALT When attempting to circumscribe a 24-sided polygon about a circle, each side coincides with the arc along a significant section

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Angolul "tangent line", ami magyarul "érintô". Az találó, mert ha osztom egy kör sugarát cos(x)-szel, egy sokszög átlójának a felét kapom. Azt szorzom sin(x)-szel — mint ha egy nagyobb kör sugara lenne —, egy érintô szakasz felét kapom. Sin / cos = tan.
Körszelet és azonos középpontú szabályos sökszög szelete

ALT Körszelet és azonos középpontú szabályos sökszög szelete

Trigonometria

ALT Trigonometria

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Ôk is egy pontos érintésben gondolkodnak, de a "tangent" elnevezés egy valós tulajdonságból ered. Szóval a magyar "érintô" kifejezés eredendôen hibás, mert a téves egy pontos érintés elméletén alapszik. Közben valójában több pontos átfedés, illetve szelés van.
Körív és egyenes vonal

ALT Körív és egyenes vonal

Körív és egyenes vonal pontjai

ALT Körív és egyenes vonal pontjai

A hagyományos, elvonatkoztatott, egy pontos érintés

ALT A hagyományos, elvonatkoztatott, egy pontos érintés

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The shape of the arc doesn't enable a single-point coincidence with a tangent line. Yet, it became a fundamental conventional concept so badly, that the Hungarian word for the tangent line reflects the single-point condition rather than the tangent property.
Arc and coinciding line.

ALT Arc and coinciding line.

Arc and coinciding line.

ALT Arc and coinciding line.

Abstract traditional concept of single-point tangency

ALT Abstract traditional concept of single-point tangency

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That becomes critical in the definition of circumscribed polygons that rely on it entirely. All those different concepts and conditions woven into the definitions make communicating them properly quite challenging for me. Basically, I'm replacing invalid concepts with valid ones.
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So, what the mathematical constant pi really stands for is the expression n×sin(180°/n)<pi<n×tan(180°/n). With n approaching infinity, that is 3.14... But that is not the true circumference to diameter ratio; it's just a failed approximate of it. ... nitter.gq/BasicGeometry/status/2…
Pi approximation

ALT Pi approximation

Pi approximation

ALT Pi approximation

Basic Geometry (@BasicGeometry) profile pictureBasic Geometry@BasicGeometryJul 25
Replying to @gmac4247
Mainstream geometry insists that a tangent line can always be drawn. A tangent line coincides with a circle only at a single point. Yet, when attempting to circumscribe a 24-sided polygon about a circle, each side coincides with the arc along a significant section.
Attempting to circumscribe a 24-sided polygon about a circle.

ALT Attempting to circumscribe a 24-sided polygon about a circle.

Zooming in on one side of an attempted 24-sided circumscribed polygon about a circle.

ALT Zooming in on one side of an attempted 24-sided circumscribed polygon about a circle.

When attempting to circumscribe a 24-sided polygon about a circle, each side coincides with the arc along a significant section

ALT When attempting to circumscribe a 24-sided polygon about a circle, each side coincides with the arc along a significant section

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That is why the pi is not — and never will be — 3.2. And that's why the pi shall never be associated with the properties of a circle. Denoting the circumference to diameter ratio as pi is just a bad convention by extremists who pretend to be much smarter than they actually are.
3.2radius² is the exact area of a circle.

ALT 3.2radius² is the exact area of a circle.

Circle and square area comparison.

ALT Circle and square area comparison.

Circle and square area comparison.

ALT Circle and square area comparison.

The joy of geometry

ALT The joy of geometry

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Don't fall for this type of fraud. The true volume of a sphere equals the cubic value of the square root of its cross-sectional area — just like a cube —, which is about a third larger than the result of this worthless, primitive, exaggerated approximate. Measure it.
Mathematica (@mathemetica) profile pictureMathematica
@mathemeticaOct 24
The sphere’s true volume is revealed by what remains after the cones are removed. Every horizontal slice of the sphere is a disk of radius √(r² − z²). The cylinder with its double cone subtracted leaves an annulus of outer radius r and inner radius |z|. Both areas equal π(r² − z²). Cavalieri’s principle therefore equates the volumes. Subtract the two cones from the cylinder: 2πr³ − 2·(⅓πr³) = ⁴⁄₃πr³. The sphere occupies exactly two-thirds of its enclosing cylinder. Geometry finds equality in the slices we choose to compare.
Image from Mathematica (@mathemetica)
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I still can't find any video proving that Archimedian "V=r³4pi/3" formula. Can you? Of course you can't, because it's about one third off. Measure it and see for yourself. nitter.gq/gmac4247/status/192125…
Gaál Sándor (@gmac4247) profile pictureGaál Sándor@gmac424710 May 2025
True geometry. Finding the volume of a sphere using liquid displacement. Prepare to be amazed! youtu.be/rXcJhFLLP0Y?si=Nxx3…
The sphere experiment 2

The sphere experiment 2

Finding the volume of a sphere via liquid displacement. Disproving...

youtube.com
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I got this latest addition wrong last week. - I confused tangent with sine, and - I calculated the left side on the double. That was presented inaccurately for almost a week. I'm sorry about that. I wanted to get it done fast, and I lost my focus. I got it fixed. Job's done.
Pi approximation

ALT Pi approximation

Gaál Sándor (@gmac4247) profile pictureGaál Sándor@gmac4247Oct 14
I finished development today. — Mostly. 🏞️🫠
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3.2radius² is the exact area of a circle. From that 3.2diameter is the exact circumference. Confirmed via measurements. The wise insist that can't be, because their theoretical circumscribed polygon limits it to about 3.14. Their polygon is not as circumscribed as they think.
3.2radius² is the exact area of a circle.

ALT 3.2radius² is the exact area of a circle.

The circumference of a coin with a diameter of 29 mm measured about 93 mm on a ruler with a thin paper strip.

ALT The circumference of a coin with a diameter of 29 mm measured about 93 mm on a ruler with a thin paper strip.

Arc and tangent line

ALT Arc and tangent line

Arc and tangent line spacing

ALT Arc and tangent line spacing

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The arc intersects the sides exactly at their quarters only if side / radius = 4 / √5.
The arc intersects the sides exactly at their quarters only if side / radius = 4 / √5.

ALT The arc intersects the sides exactly at their quarters only if side / radius = 4 / √5.

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Angolul írtam mindent a mértanról — kivéve néhány szakkifejezést, amire az angolban is latin, vagy mit tudom én milyen szót használnak —, mert a célközönség nemzetközi és mostanság a mûszaki irodalom nemzetközi nyelve jellemzôen az angol. Vannak egész jó fordító alkalmazások.
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Tavaly le is írtam a nagyját magyarul, de azóta volt amit átfogalmaztam. A lényeg, hogy egy darabig nem lesz igazán jó magyar nyelvû szöveg. De nem is kell. Az ábrák és az egyenletek a lényeg. Nem író vagyok. A hagyományos mértan is a szövegnél ment félre.
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Vannak akik úgy állítják be, mint ha a jól megfogalmazott szöveg valami tekintélyt parancsolna, vagy ilyesmi. Ettôl nagyobbat nem is tévedhetnének. A mértanra sokkal inkább jellemzô az a mondás, hogy egy kép felér ezer szóval. Talán egyszer leírom magyarul is, de nem mostanában.
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