# Heart to heart with 3D Math

I have studied mathematics and worked with Mathematica for quite some time , as can be seen from my problem solvers page and the science books page .

Some curves and surfaces have nice 3 D heart shapes when plotted the right way .Maybe it’s Valentine’s day by coincidence and maybe not  , anyway here are some examples of ‘math’ hearts with Mathematica ( and sometimes a little Photoshop):

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For those interested in math and coding , I’ll give as an example the Mathematica code for the big red heart with white background  :

And here are some more (customized ) heart surfaces. Shows  there is beauty in mathematics .

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The ‘plump’ red heart with the orange background is a heart-shaped surface given by the Taubin equation:

And also by the equation :

The Mathematica code for the Taubin heart-shaped surface is:

# Some more thoughts about education

Children who have the possibility to finish school at an early age and to start their higher education at a very young age are often called child prodigies or gifted children.They are
frequently treated as curiosities or rare people with an acute intelligence and uncommon abilities .These kids may have special aptitudes but there is a plausible explanation for their situation .To put it simply , either they have fast stages of intellectual development and they were noticed and helped by their parents , family, teachers and/or professionals in order to have an accelerated education , skip grades and go to college at a very young age (probably 9 or 11 or 12) , or they have  average (or slightly higher than average) developmental stages and were also taught , helped and trained by their  parents , teachers and education professionals in ways which allowed them to finish school early and to enter the university precociously.

I think  the way these precocious or gifted children are taught ,  accelerated education and the opportunity to complete one’s studies early and to start university studies at a younger age ought to be given and extended progressively to all children .This way fast learners will not be left behind or neglected , and generation after generation more and more kids and young people will have faster stages of growth and will be able to assimilate more information at younger ages . When lowering the age of entry to the university , some youngsters could  start higher education at the age of 11 or 13 , but the minimum age could be set at 10 or 9 .

Moreover , a smaller , restricted acceleration could be applied  to higher education and university studies . For example , becoming an engineer requires four years of study in certain countries and five years in other countries . The five-year programs could be condensed into four years by appropriate methods such as adding hours to the four years of study , condensing some courses and/or adding summer courses . Medical studies usually take eight or ten years or even longer to be completed . By applying the convenient acceleration and condensation of courses and studies the eight years of study could be reduced to seven and the ten to nine and so forth , without loss of knowledge or qualification for the future doctor or medical practitioner. Similarly predoctoral and master’s degree studies could be reduced from five to four years or from six years to five.

Hence in the long run everybody would benefit from this reform and acceleration of education , and the gap between youngsters considered to be average or normal and child prodigies would be less wide ,  which I consider to be  a good thing , providing more educational equality and efficiency .

A short remark concerning one aspect of education: Nowadays young people are generally becoming sexually aware (and sometimes active) at younger ages compared with older generations , and this fact should be taken into consideration in education.

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I’ll add a final remark here : The general system  of education followed in a large number of countries today was formed about two or three centuries ago . In a country such as France  a major reform in education took place , coinciding with the period of the French Revolution , and with the  time when the first  successful endoatmospheric human flights were made using
hot air balloons  . One thing to note is that the Montgolfier brothers , who flew the first balloon , were elevated to the nobility as a reward . Nowadays the old aristocratic nobility
titles are gone in France , so people are instead rewarded by  honorary degrees ,which can be considered as  contemporary nobility titles . Many of the thinkers and scientists who
contributed to the reform of  the education system in France two centuries ago , such as Condorcet and Laplace , had acquired old nobility  titles (such as ‘marquis’) , titles related
to the ‘Ancien Régime’ , and were forming the new system of instruction which would provide the new (education related)  nobility titles in France and elsewhere . Perhaps nobility
titles in one form or another will frequently or regularly succeed one another with time , but there should always be room for reassessment , open-mindedness , change and reform . Wilhelm von Humboldt witnessed and was influenced by the French Revolution . He attempted to reform and reorganize the Prussian and German educational system , and founded the University of Berlin in 1810 . In the following decades of the nineteenth century , educators in the United states and other countries emulated and implemented the Prussian education system.
The rest of the  nineteenth and most of the twentieth centuries witnessed generally lesser scattered reforms in education , the elaboration of educational and psychological theories , and were the scene of  a progressive worldwide adoption of the educational ideas and institutions developed in European nations before , during and after the Enlightenment period and the  French Revolution .This epoch  coincided with the development of aerodynamics,
aeronautics and heavier-than-air aircraft , and with the beginnings of astronautics and space exploration .

The following idea is worth considering: While the existent educational system , with its diplomas and degrees and its requirements to study or act in a certain way and to start
higher education at the average age of eighteen , has worked well and has been sufficient for people who move or travel on planet Earth and in its surroundings, I think the reform of
the educational system  I have proposed and written about here , i.e accelerating education , skipping  or condensing  grades and lowering the age of entry to college or the university , will
prove to be important , more efficient  , and necessary in this era of planetary globalization , and more so  in the future , as humans in the age of space travel  intend or attempt to go to Mars and  others planets , traveling in the solar system beyond the Earth-Moon system and the immediate vicinity of planet Earth.

The notion of educational level is usually related to the educational system or framework within which it is used . For example , there are three cycles (sometimes called ‘levels’) in
higher education . Levels are considered as corresponding to diplomas and degrees ( i. e. High School diploma , Bachelor’s , Master’s degree , PhD) awarded by the educational
establishments , so that people with higher degrees are considered to have a better or higher level . Sometimes the word level is also used (especially in selective schools or
educational institutions) in the sense that students with higher grades or marks have a higher or better level than those with lower or average grades.

It has been shown over the years and decades   by numerous success stories that students with excellent grades ( ‘A’ students in the American education system) don’t necessarily
achieve better in life than average (or ‘C’) students . Not to mention the many  examples of college dropouts who were successful and made important , notable achievements
compared with other people who finished their higher education and got university degrees but accomplished less.

I have written in earlier posts  about kids or young people who have  fast stages , states or phases of  intellectual development and are often misunderstood , cannot adapt to the existing education system and even  fail in it.These young people may   also be called fast learners or fast receivers (of information or knowledge) . I think one possible explanation for the
achievements of average students or dropouts as compared with ‘A’ students  is that a number of these average students and dropouts are fast learners . Their personal  stages ( or
cycles ) of development are faster than and incompatible with the stages or cycles acknowledged by the educational system . Therefore they don’t follow the system well and don’t study as required by the current curricula  , and tend to fail or leave the system  altogether.

Sometimes people pass through a period of time (probably a few years) during which they study a lot and have high grades , then this period ends and they start reading a lot
instead of studying and have lower or average grades . This period takes place at a younger age for some people and ends earlier for them , while it can last longer for others . That’s
why there are people who have high grades at school and then become average ‘C’ ( or even ‘D’) students at the university , whereas others go on and have high grades for a long time
and can continue studying and having degrees till they are thirty or forty years of age . The fast  learners in these cases are the ones who stop studying a lot or stop succeeding in the
education system at a younger age .

It can be seen by the analysis I’ve made above that ‘level’ and ‘educational level’ are expressions that are used abusively in relation to the current educational system . They can lead
to discriminatory attitudes and do not reflect the real potentials , aptitudes or abilities of students and young people.

In the third cycle of higher education it is usually not required of students to study a lot but to read a lot and to do personal research for about three years in order to prepare and
obtain their PhD degree . This is one proof that in the existing system the final , more advanced stage or cycle takes place when reading and research , not studying and getting good grades ,
become the essential requirements , and although this is commonly expected from students at ages over 20 or 25 or 30 , fast learners reach  this stage at younger ages , maybe at 18 or 15  or
younger . Therefore accelerating  education and lowering the age of entry to the university is necessary to take into account the fast ones and their abilities.

Students and pupils who have high grades or a high academic achievement generally get certain privileges and are accepted before others as they pass the  entrance examinations of high
schools or universities . They are  preferred by cram schools and schools which apply academic selection. In a country such as France an example of selective schools would be
the ‘Classes préparatoires aux grandes écoles’  , and in England the grammar schools.
By the arguments stated above , and since this period of having high marks is one that lasts for a certain time and then passes away ,
it would be preferable if people with high grades are given less privileges , the important thing being to pass the exams and tests and succeed in the educational system ,even within a reformed system where educational acceleration is applied .

When or if (young) people have good grades and high academic achievement , then good for them , they can be proud of what they
have done but can also keep it for themselves . I remember I was the top of my class at school and I had high grades for about seven years before the age of fifteen , and I was at that
time very proud of having such very good results , but then things changed and I began reading more than studying and started having average grades . So one could be happy
about one’s good grades and tell others about it , but there is no need to brag a lot about it and no need to be given additional privileges.