ºÝºÝߣshows by User: mikhajduk / http://www.slideshare.net/images/logo.gif ºÝºÝߣshows by User: mikhajduk / Thu, 21 Jan 2016 18:19:19 GMT ºÝºÝߣShare feed for ºÝºÝߣshows by User: mikhajduk Ptolemy's theorem visualisation. 3D graphics. /slideshow/ptolemys-theorem-visualisation-3d-graphics/57338468 ptolemystheorem-160121181919
Ptolemy's theorem states the following: a convex quadrilateral can be inscribed in a circle if and only if the product of the lengths of one pair of opposite sides added to the product of the lengths of the other pair is equal to the product of the lengths of the diagonals. Thus, in a cyclic quadrilateral ABCD we have AB*DC + AD*BC = AC*BD ]]>
Ptolemy's theorem states the following: a convex quadrilateral can be inscribed in a circle if and only if the product of the lengths of one pair of opposite sides added to the product of the lengths of the other pair is equal to the product of the lengths of the diagonals. Thus, in a cyclic quadrilateral ABCD we have AB*DC + AD*BC = AC*BD ]]> Thu, 21 Jan 2016 18:19:19 GMT /slideshow/ptolemys-theorem-visualisation-3d-graphics/57338468 mikhajduk@slideshare.net(mikhajduk) Ptolemy's theorem visualisation. 3D graphics. mikhajduk Ptolemy's theorem states the following: a convex quadrilateral can be inscribed in a circle if and only if the product of the lengths of one pair of opposite sides added to the product of the lengths of the other pair is equal to the product of the lengths of the diagonals. Thus, in a cyclic quadrilateral ABCD we have AB*DC + AD*BC = AC*BD <img style="border:1px solid #C3E6D8;float:right;" alt="" src="https://cdn.slidesharecdn.com/ss_thumbnails/ptolemystheorem-160121181919-thumbnail.jpg?width=120&amp;height=120&amp;fit=bounds" /><br> Ptolemy&#39;s theorem states the following: a convex quadrilateral can be inscribed in a circle if and only if the product of the lengths of one pair of opposite sides added to the product of the lengths of the other pair is equal to the product of the lengths of the diagonals. Thus, in a cyclic quadrilateral ABCD we have AB*DC + AD*BC = AC*BD
Ptolemy's theorem visualisation. 3D graphics. from MikoナＢj Hajduk
]]> 188 4 https://cdn.slidesharecdn.com/ss_thumbnails/ptolemystheorem-160121181919-thumbnail.jpg?width=120&height=120&fit=bounds presentation Black http://activitystrea.ms/schema/1.0/post http://activitystrea.ms/schema/1.0/posted 0 An inequality painted on the vase body. /slideshow/an-inequality-painted-on-the-vase-body/57286424 vaseinequality-160120165259
The 3D picture shows an attempt at combining applied art with pure mathematics. The vase shown in the picture is an example of ceramic vessel that could exist in real world.]]>
The 3D picture shows an attempt at combining applied art with pure mathematics. The vase shown in the picture is an example of ceramic vessel that could exist in real world.]]> Wed, 20 Jan 2016 16:52:59 GMT /slideshow/an-inequality-painted-on-the-vase-body/57286424 mikhajduk@slideshare.net(mikhajduk) An inequality painted on the vase body. mikhajduk The 3D picture shows an attempt at combining applied art with pure mathematics. The vase shown in the picture is an example of ceramic vessel that could exist in real world. <img style="border:1px solid #C3E6D8;float:right;" alt="" src="https://cdn.slidesharecdn.com/ss_thumbnails/vaseinequality-160120165259-thumbnail.jpg?width=120&amp;height=120&amp;fit=bounds" /><br> The 3D picture shows an attempt at combining applied art with pure mathematics. The vase shown in the picture is an example of ceramic vessel that could exist in real world.
An inequality painted on the vase body. from MikoナＢj Hajduk
]]> 101 4 https://cdn.slidesharecdn.com/ss_thumbnails/vaseinequality-160120165259-thumbnail.jpg?width=120&height=120&fit=bounds presentation Black http://activitystrea.ms/schema/1.0/post http://activitystrea.ms/schema/1.0/posted 0 Still life with vases - a 3D visualisation. /slideshow/still-life-with-vases-a-3d-visualisation/57285587 stilllifewithvases-160120163328
The picture presenting a still life with two vases made of black polished ceramics, standing on blocks of granite and limestone, placed in an empty room illuminated by sunlight going through a big window.]]>
The picture presenting a still life with two vases made of black polished ceramics, standing on blocks of granite and limestone, placed in an empty room illuminated by sunlight going through a big window.]]> Wed, 20 Jan 2016 16:33:28 GMT /slideshow/still-life-with-vases-a-3d-visualisation/57285587 mikhajduk@slideshare.net(mikhajduk) Still life with vases - a 3D visualisation. mikhajduk The picture presenting a still life with two vases made of black polished ceramics, standing on blocks of granite and limestone, placed in an empty room illuminated by sunlight going through a big window. <img style="border:1px solid #C3E6D8;float:right;" alt="" src="https://cdn.slidesharecdn.com/ss_thumbnails/stilllifewithvases-160120163328-thumbnail.jpg?width=120&amp;height=120&amp;fit=bounds" /><br> The picture presenting a still life with two vases made of black polished ceramics, standing on blocks of granite and limestone, placed in an empty room illuminated by sunlight going through a big window.
Still life with vases - a 3D visualisation. from MikoナＢj Hajduk
]]> 141 5 https://cdn.slidesharecdn.com/ss_thumbnails/stilllifewithvases-160120163328-thumbnail.jpg?width=120&height=120&fit=bounds presentation Black http://activitystrea.ms/schema/1.0/post http://activitystrea.ms/schema/1.0/posted 0 How to draw arcs of huge circles - usage of the "drawing tool". /slideshow/how-to-draw-arcs-of-huge-circles-usage-of-the-drawing-tool/57285112 woodenblockusage-160120162324
The slide explaining visually the procedure of drawing huge circles. A theoretical background of the method has been described in another slide.]]>
The slide explaining visually the procedure of drawing huge circles. A theoretical background of the method has been described in another slide.]]> Wed, 20 Jan 2016 16:23:24 GMT /slideshow/how-to-draw-arcs-of-huge-circles-usage-of-the-drawing-tool/57285112 mikhajduk@slideshare.net(mikhajduk) How to draw arcs of huge circles - usage of the "drawing tool". mikhajduk The slide explaining visually the procedure of drawing huge circles. A theoretical background of the method has been described in another slide. <img style="border:1px solid #C3E6D8;float:right;" alt="" src="https://cdn.slidesharecdn.com/ss_thumbnails/woodenblockusage-160120162324-thumbnail.jpg?width=120&amp;height=120&amp;fit=bounds" /><br> The slide explaining visually the procedure of drawing huge circles. A theoretical background of the method has been described in another slide.
How to draw arcs of huge circles - usage of the "drawing tool". from MikoナＢj Hajduk
]]> 115 4 https://cdn.slidesharecdn.com/ss_thumbnails/woodenblockusage-160120162324-thumbnail.jpg?width=120&height=120&fit=bounds presentation Black http://activitystrea.ms/schema/1.0/post http://activitystrea.ms/schema/1.0/posted 0 How to draw arcs of huge circles - description. /mikhajduk/how-to-draw-arcs-of-huge-circles-description woodenblockdescription-160120133534
In case you are interested in drawing circles with a huge diameter (by "huge" I mean here really big ones, measured in kilometers or so) there is a tricky idea needed because the rope-and-the-stick method won't work for circles with a diameter bigger than several dozen meters. Here you can use a method based on the well known fact from geometry stating that all angles inscribed in a circle and subtended by the same chord (lying on the same side of the chord) are equal.]]>
In case you are interested in drawing circles with a huge diameter (by "huge" I mean here really big ones, measured in kilometers or so) there is a tricky idea needed because the rope-and-the-stick method won't work for circles with a diameter bigger than several dozen meters. Here you can use a method based on the well known fact from geometry stating that all angles inscribed in a circle and subtended by the same chord (lying on the same side of the chord) are equal.]]> Wed, 20 Jan 2016 13:35:34 GMT /mikhajduk/how-to-draw-arcs-of-huge-circles-description mikhajduk@slideshare.net(mikhajduk) How to draw arcs of huge circles - description. mikhajduk In case you are interested in drawing circles with a huge diameter (by "huge" I mean here really big ones, measured in kilometers or so) there is a tricky idea needed because the rope-and-the-stick method won't work for circles with a diameter bigger than several dozen meters. Here you can use a method based on the well known fact from geometry stating that all angles inscribed in a circle and subtended by the same chord (lying on the same side of the chord) are equal. <img style="border:1px solid #C3E6D8;float:right;" alt="" src="https://cdn.slidesharecdn.com/ss_thumbnails/woodenblockdescription-160120133534-thumbnail.jpg?width=120&amp;height=120&amp;fit=bounds" /><br> In case you are interested in drawing circles with a huge diameter (by &quot;huge&quot; I mean here really big ones, measured in kilometers or so) there is a tricky idea needed because the rope-and-the-stick method won&#39;t work for circles with a diameter bigger than several dozen meters. Here you can use a method based on the well known fact from geometry stating that all angles inscribed in a circle and subtended by the same chord (lying on the same side of the chord) are equal.
How to draw arcs of huge circles - description. from MikoナＢj Hajduk
]]> 128 4 https://cdn.slidesharecdn.com/ss_thumbnails/woodenblockdescription-160120133534-thumbnail.jpg?width=120&height=120&fit=bounds presentation Black http://activitystrea.ms/schema/1.0/post http://activitystrea.ms/schema/1.0/posted 0 Calculation of the volume of a bottle partially filled with a fluid. /slideshow/calculation-of-the-volume-of-a-bottle-partially-filled-with-a-fluid/57276337 bottlesslideshare-160120132606
How to calculate the volume of a flat-bottomed, corked bottle, partially filled with a fluid having only the ruler as a measuring tool? The 3D graphics made by me and presented below gives an answer to this question.]]>
How to calculate the volume of a flat-bottomed, corked bottle, partially filled with a fluid having only the ruler as a measuring tool? The 3D graphics made by me and presented below gives an answer to this question.]]> Wed, 20 Jan 2016 13:26:05 GMT /slideshow/calculation-of-the-volume-of-a-bottle-partially-filled-with-a-fluid/57276337 mikhajduk@slideshare.net(mikhajduk) Calculation of the volume of a bottle partially filled with a fluid. mikhajduk How to calculate the volume of a flat-bottomed, corked bottle, partially filled with a fluid having only the ruler as a measuring tool? The 3D graphics made by me and presented below gives an answer to this question. <img style="border:1px solid #C3E6D8;float:right;" alt="" src="https://cdn.slidesharecdn.com/ss_thumbnails/bottlesslideshare-160120132606-thumbnail.jpg?width=120&amp;height=120&amp;fit=bounds" /><br> How to calculate the volume of a flat-bottomed, corked bottle, partially filled with a fluid having only the ruler as a measuring tool? The 3D graphics made by me and presented below gives an answer to this question.
Calculation of the volume of a bottle partially filled with a fluid. from MikoナＢj Hajduk
]]> 179 4 https://cdn.slidesharecdn.com/ss_thumbnails/bottlesslideshare-160120132606-thumbnail.jpg?width=120&height=120&fit=bounds presentation Black http://activitystrea.ms/schema/1.0/post http://activitystrea.ms/schema/1.0/posted 0 Permutation theorem and its use to proving inequalities. /slideshow/permutation-theorem-and-its-use-to-proving-inequalities/53902672 permutationtheorem-151013234839-lva1-app6891
The permutation theorem is very useful when dealing with inequalities between sums of products of the two real number sequences. In numerous cases inequalities otherwise difficult to prove can be proven almost automatically.]]>
The permutation theorem is very useful when dealing with inequalities between sums of products of the two real number sequences. In numerous cases inequalities otherwise difficult to prove can be proven almost automatically.]]> Tue, 13 Oct 2015 23:48:39 GMT /slideshow/permutation-theorem-and-its-use-to-proving-inequalities/53902672 mikhajduk@slideshare.net(mikhajduk) Permutation theorem and its use to proving inequalities. mikhajduk The permutation theorem is very useful when dealing with inequalities between sums of products of the two real number sequences. In numerous cases inequalities otherwise difficult to prove can be proven almost automatically. <img style="border:1px solid #C3E6D8;float:right;" alt="" src="https://cdn.slidesharecdn.com/ss_thumbnails/permutationtheorem-151013234839-lva1-app6891-thumbnail.jpg?width=120&amp;height=120&amp;fit=bounds" /><br> The permutation theorem is very useful when dealing with inequalities between sums of products of the two real number sequences. In numerous cases inequalities otherwise difficult to prove can be proven almost automatically.
Permutation theorem and its use to proving inequalities. from MikoナＢj Hajduk
]]> 106 4 https://cdn.slidesharecdn.com/ss_thumbnails/permutationtheorem-151013234839-lva1-app6891-thumbnail.jpg?width=120&height=120&fit=bounds presentation Black http://activitystrea.ms/schema/1.0/post http://activitystrea.ms/schema/1.0/posted 0 The sum of the triangle sides lengths reciprocals vs a cyclic sum of a specific form. /slideshow/the-sum-of-the-triangle-sides-lengths-reciprocals-vs-a-cyclic-sum-of-a-specific-form/53839371 trianglesidesineqproof-151012173705-lva1-app6892
Proof of the inequality between the sum of the reciprocals of a triangle sides lengths and a cyclic sum of a specific form. Use of a transformed inequality between the arithmetic mean and the harmonic mean. ]]>
Proof of the inequality between the sum of the reciprocals of a triangle sides lengths and a cyclic sum of a specific form. Use of a transformed inequality between the arithmetic mean and the harmonic mean. ]]> Mon, 12 Oct 2015 17:37:05 GMT /slideshow/the-sum-of-the-triangle-sides-lengths-reciprocals-vs-a-cyclic-sum-of-a-specific-form/53839371 mikhajduk@slideshare.net(mikhajduk) The sum of the triangle sides lengths reciprocals vs a cyclic sum of a specific form. mikhajduk Proof of the inequality between the sum of the reciprocals of a triangle sides lengths and a cyclic sum of a specific form. Use of a transformed inequality between the arithmetic mean and the harmonic mean. <img style="border:1px solid #C3E6D8;float:right;" alt="" src="https://cdn.slidesharecdn.com/ss_thumbnails/trianglesidesineqproof-151012173705-lva1-app6892-thumbnail.jpg?width=120&amp;height=120&amp;fit=bounds" /><br> Proof of the inequality between the sum of the reciprocals of a triangle sides lengths and a cyclic sum of a specific form. Use of a transformed inequality between the arithmetic mean and the harmonic mean.
The sum of the triangle sides lengths reciprocals vs a cyclic sum of a specific form. from MikoナＢj Hajduk
]]> 226 4 https://cdn.slidesharecdn.com/ss_thumbnails/trianglesidesineqproof-151012173705-lva1-app6892-thumbnail.jpg?width=120&height=120&fit=bounds presentation Black http://activitystrea.ms/schema/1.0/post http://activitystrea.ms/schema/1.0/posted 0 Complex Integral /slideshow/complex-integral/52870680 complexintegralfb-150917003048-lva1-app6892
Evaluation of integrals of the given functions along the unit circle on the complex plane. Application of the parametrization method. Evaluation of the integral of an odd function.]]>
Evaluation of integrals of the given functions along the unit circle on the complex plane. Application of the parametrization method. Evaluation of the integral of an odd function.]]> Thu, 17 Sep 2015 00:30:48 GMT /slideshow/complex-integral/52870680 mikhajduk@slideshare.net(mikhajduk) Complex Integral mikhajduk Evaluation of integrals of the given functions along the unit circle on the complex plane. Application of the parametrization method. Evaluation of the integral of an odd function. <img style="border:1px solid #C3E6D8;float:right;" alt="" src="https://cdn.slidesharecdn.com/ss_thumbnails/complexintegralfb-150917003048-lva1-app6892-thumbnail.jpg?width=120&amp;height=120&amp;fit=bounds" /><br> Evaluation of integrals of the given functions along the unit circle on the complex plane. Application of the parametrization method. Evaluation of the integral of an odd function.
Complex Integral from MikoナＢj Hajduk
]]> 1085 4 https://cdn.slidesharecdn.com/ss_thumbnails/complexintegralfb-150917003048-lva1-app6892-thumbnail.jpg?width=120&height=120&fit=bounds presentation Black http://activitystrea.ms/schema/1.0/post http://activitystrea.ms/schema/1.0/posted 0 Indescribable numbers /slideshow/indescribable-numbers/51633621 indescribable-150814150439-lva1-app6892
A few digressions on the theme of indescribable numbers. Proof of the fact that there exist real numbers that may be indescribable, i.e. impossible to express in any language based on the finite alphabet of symbols.]]>
A few digressions on the theme of indescribable numbers. Proof of the fact that there exist real numbers that may be indescribable, i.e. impossible to express in any language based on the finite alphabet of symbols.]]> Fri, 14 Aug 2015 15:04:38 GMT /slideshow/indescribable-numbers/51633621 mikhajduk@slideshare.net(mikhajduk) Indescribable numbers mikhajduk A few digressions on the theme of indescribable numbers. Proof of the fact that there exist real numbers that may be indescribable, i.e. impossible to express in any language based on the finite alphabet of symbols. <img style="border:1px solid #C3E6D8;float:right;" alt="" src="https://cdn.slidesharecdn.com/ss_thumbnails/indescribable-150814150439-lva1-app6892-thumbnail.jpg?width=120&amp;height=120&amp;fit=bounds" /><br> A few digressions on the theme of indescribable numbers. Proof of the fact that there exist real numbers that may be indescribable, i.e. impossible to express in any language based on the finite alphabet of symbols.
Indescribable numbers from MikoナＢj Hajduk
]]> 174 5 https://cdn.slidesharecdn.com/ss_thumbnails/indescribable-150814150439-lva1-app6892-thumbnail.jpg?width=120&height=120&fit=bounds presentation Black http://activitystrea.ms/schema/1.0/post http://activitystrea.ms/schema/1.0/posted 0 Late Spring 2015 Photos /mikhajduk/late-spring2015 latespring2015-150623192337-lva1-app6892
Photos taken in May and June 2015.]]>
Photos taken in May and June 2015.]]> Tue, 23 Jun 2015 19:23:37 GMT /mikhajduk/late-spring2015 mikhajduk@slideshare.net(mikhajduk) Late Spring 2015 Photos mikhajduk Photos taken in May and June 2015. <img style="border:1px solid #C3E6D8;float:right;" alt="" src="https://cdn.slidesharecdn.com/ss_thumbnails/latespring2015-150623192337-lva1-app6892-thumbnail.jpg?width=120&amp;height=120&amp;fit=bounds" /><br> Photos taken in May and June 2015.
Late Spring 2015 Photos from MikoナＢj Hajduk
]]> 143 1 https://cdn.slidesharecdn.com/ss_thumbnails/latespring2015-150623192337-lva1-app6892-thumbnail.jpg?width=120&height=120&fit=bounds presentation Black http://activitystrea.ms/schema/1.0/post http://activitystrea.ms/schema/1.0/posted 0 Cube root /slideshow/cube-root-49631334/49631334 cuberoot-150620125338-lva1-app6891
A mathematical explanation of a simple trick that shows how to quickly calculate the cube root of a maximally 6-digit number that was given by a spectator. The spectator chooses a natural two-digit number, keeps it in secret and only information he/she gives publicly is its cube. Conjurer's task is to quickly guess the original number.]]>
A mathematical explanation of a simple trick that shows how to quickly calculate the cube root of a maximally 6-digit number that was given by a spectator. The spectator chooses a natural two-digit number, keeps it in secret and only information he/she gives publicly is its cube. Conjurer's task is to quickly guess the original number.]]> Sat, 20 Jun 2015 12:53:37 GMT /slideshow/cube-root-49631334/49631334 mikhajduk@slideshare.net(mikhajduk) Cube root mikhajduk A mathematical explanation of a simple trick that shows how to quickly calculate the cube root of a maximally 6-digit number that was given by a spectator. The spectator chooses a natural two-digit number, keeps it in secret and only information he/she gives publicly is its cube. Conjurer's task is to quickly guess the original number. <img style="border:1px solid #C3E6D8;float:right;" alt="" src="https://cdn.slidesharecdn.com/ss_thumbnails/cuberoot-150620125338-lva1-app6891-thumbnail.jpg?width=120&amp;height=120&amp;fit=bounds" /><br> A mathematical explanation of a simple trick that shows how to quickly calculate the cube root of a maximally 6-digit number that was given by a spectator. The spectator chooses a natural two-digit number, keeps it in secret and only information he/she gives publicly is its cube. Conjurer&#39;s task is to quickly guess the original number.
Cube root from MikoナＢj Hajduk
]]> 410 1 https://cdn.slidesharecdn.com/ss_thumbnails/cuberoot-150620125338-lva1-app6891-thumbnail.jpg?width=120&height=120&fit=bounds presentation Black http://activitystrea.ms/schema/1.0/post http://activitystrea.ms/schema/1.0/posted 0 https://cdn.slidesharecdn.com/profile-photo-mikhajduk-48x48.jpg?cb=1523766264 mikhajduk.houa.org https://cdn.slidesharecdn.com/ss_thumbnails/ptolemystheorem-160121181919-thumbnail.jpg?width=320&height=320&fit=bounds slideshow/ptolemys-theorem-visualisation-3d-graphics/57338468 Ptolemy&#39;s theorem visu... https://cdn.slidesharecdn.com/ss_thumbnails/vaseinequality-160120165259-thumbnail.jpg?width=320&height=320&fit=bounds slideshow/an-inequality-painted-on-the-vase-body/57286424 An inequality painted ... https://cdn.slidesharecdn.com/ss_thumbnails/stilllifewithvases-160120163328-thumbnail.jpg?width=320&height=320&fit=bounds slideshow/still-life-with-vases-a-3d-visualisation/57285587 Still life with vases ...