Handshakes

Warm up sketch inspired by the naked-geometry.

On geogebratube for you to play. Has a tool to make multiple copies of your own.

Handshakes

Warm up sketch inspired by the naked-geometry.

On geogebratube for you to play. Has a tool to make multiple copies of your own.

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loversdreamersandyou posted a still from this bit this morning and I reblogged it on geekhombre. Turns out it’s from a pretty sophisticated math bit. AND it’s the last skit Jim Henson and Frank Oz did together on Sesame Street.

Type of Spirals:A spiral is a curve in the plane or in the space, which runs around a centre in a special way.Different spirals follow. Most of them are produced by formulas:The radius r(t) and the angle t are proportional for the simplest spiral, the spiral of Archimedes. Therefore the equation is:

(3) Polar equation: r(t) = at [a is constant].

From this follows

(2) Parameter form: x(t) = at cos(t), y(t) = at sin(t),

(1) Central equation: x²+y² = a²[arc tan (y/x)]².You can make a spiral by two motions of a point: There is a uniform motion in a fixed direction and a motion in a circle with constant speed. Both motions start at the same point.

(1) The uniform motion on the left moves a point to the right. - There are nine snapshots.

(2) The motion with a constant angular velocity moves the point on a spiral at the same time. - There is a point every 8th turn.

(3) A spiral as a curve comes, if you draw the point at every turn(Image).

Figure 1: (1)Archimedean spiral -(2)Equiangular Spiral (Logarithmic Spiral, Bernoulli’s Spiral).Figure 2 : (1)Clothoide (Cornu Spiral) -(2)Golden spiral (Fibonacci number).More Spirals: If you replace the term r(t)=at of the Archimedean spiral by other terms, you get a number of new spirals. There are six spirals, which you can describe with the functions f(x)=x^a [a=2,1/2,-1/2,-1] and f(x)=exp(x), f(x)=ln(x). You distinguish two groups depending on how the parameter t grows from 0.

Figure 4:If the absolute modulus of a function r(t) is increasing, the spirals run from inside to outside and go above all limits. The spiral 1 is called parabolic spiral or Fermat’s spiral.Figure 5:If the absolute modulus of a function r(t) is decreasing, the spirals run from outside to inside. They generally run to the centre, but they don’t reach it. There is a pole. Spiral 2 is called the Lituus (crooked staff).

Figure 7:Spirals Made of Line Segments.

Source:Spirals by Jürgen Köller.See more on Wikipedia: Spiral, Archimedean spiral, Cornu spiral, Fermat’s spiral, Hyperbolic spiral, Lituus, Logarithmic spiral,

Fibonacci spiral, Golden spiral, Rhumb line, Ulam spiral,

Hermann Heights Monument, Hermannsdenkmal.

Image:I shared at Spirals by Jürgen Köller - Ferns by Margaret Oomen & Ferns by Rocky.

Spiral compulsion. But this is a handy reference.

∞ the 1st ever naked geometry piece, 2003! ∞ tumblr ∞ facebook ∞ etsy ∞

I might call this the handshake spiral.

A real Treasure Trove! :) From total beginner to more advanced.

Wow! Simon Gregg got playing with the same 20-80-80 triangle as I did, but his results are a lot prettier! Read the post. Wonder at the marvels.

David Marain (@dmarain) shared an angle chasing puzzle in the 80-80-20 isosceles triangle. Quite a nice one.

I chased a bit, then modeled in GeoGebra. Then I made a tool for making an ASA triangle which was handy for making angle chasing problems. Then I generalized the problem, and found that the 80-80-20 triangle had lots of neat situations that other triangles do not. (Here it is on GGBTube)

Is there something that makes this triangle special?

P.S. Turns out Simon Gregg (@Simon_Gregg) was also intrigued by this, and made some great mathart along the way. Check out his post.

Opere di Eugenio Carmi.

Opere esposte a Spoleto (PG) Italy.

http://www.spoletoarte.it/art_Eugenio_Carmi.php (10 foto)

wonderful.

Two curves cut all circles at right angles: straight line and a tractrix.

I didn’t know what a tractrix was, so I Googled it! Tractrix definition from Wikipedia:

Tractrix(from the Latin verbtrahere”pull, drag”; plural:tractrices) is the curve along which an object moves, under the influence of friction, when pulled on a horizontal plane by a line segment attached to a tractor (pulling) point that moves at a right angle to the initial line between the object and the puller at an infinitesimal speed.Here’s a gif from Wikipedia showing a tractrix being created from dragging a pole: