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    Nature, The Golden Ratio,
    and Fibonacci too ...

    Plants can grow new cells in spirals, such as the pattern of seeds in this beautiful sunflower.

    The spiral happens naturally because each new cell is formed after a turn.

    "New cell, then turn,
    then another cell, then turn, ..."

    How Far to Turn?

    So, if you were a plant, how much of a turn would you have in between new cells?

    If you don't turn at all, you get a straight line.

    But that is a very poor design ... you want something round that will hold together with no gaps.

    Why not try to find the best value for yourself?

    Try different values, like 0.75, 0.9, 3.1416, 0.62, etc.

    Remember, you are trying to make a pattern with no gaps from start to end:

    (By the way, it doesn't matter about the whole number part, like 1. or 5. because they are full revolutions that point us back in the same direction.)

    What Did You Get?

    If you got something that ends like 0.618 (or 0.382, which is 1-0.618) then "Congratulations, you are a successful member of the plant kingdom!"

    That is because the Golden Ratio (1.61803...) is the best solution, and the Sunflower has found this out in its own natural way.

    Try it ... it should look like this.


    Any number that is a simple fraction (example: 0.75 is 3/4, and 0.95 is 19/20, etc) will, after a while, make a pattern of lines stacking up, which makes gaps.

    But the Golden Ratio (its symbol is the Greek letter Phi, shown at left) is an expert at not being any fraction.

    It is an Irrational Number (meaning we cannot write it as a simple fraction), but more than that ... it is as far as we can get from being near any fraction.

    Just being irrational is not enough

    Pi (3.141592654...), which is also irrational.

    Unfortunately it has a decimal very close to 1/7 0.142857...), so it ends up with 7 arms.

    e (2.71828...) also irrational, does not work either because its decimal is close to 5/7 (0.714285...), so it also ends up with 7 arms.

    So, How Does the Golden Ratio Work?

    One of the special properties of the Golden Ratio is that it can be defined in terms of itself, like this:

    (In numbers: 1.61803... = 1 + 1/1.61803...)

    That can be expanded into this fraction that goes on for ever (called a "continued fraction"):

    So, it neatly slips in between simple fractions.

    Fibonacci Numbers

    There is a special relationship between the Golden Ratio and Fibonacci Numbers (0, 1, 1, 2, 3, 5, 8, 13, 21, ... etc, each number is the sum of the two numbers before it).

    When we take any two successive (one after the other) Fibonacci Numbers, their ratio is very close to the Golden Ratio:



    B / A



















    So, just like we naturally get seven arms when we use 0.142857 (1/7), we tend to get Fibonacci Numbers when we use the Golden Ratio.

    Try counting the spiral arms - the "left turning" spirals, and then the "right turning" spirals ... what numbers did you get?

    Spiral Leaf Growth

    This interesting behavior is not just found in sunflower seeds.

    Leaves, branches and petals can grow in spirals, too.

    Why? So that new leaves don't block the sun from older leaves, or so that the maximum amount of rain or dew gets directed down to the roots.

    In fact, when a plant has spirals the rotation tends to be a fraction made with two successive (one after the other) Fibonacci Numbers, for example:
    A half rotation is 1/2 (1 and 2 are Fibonacci Numbers)
    3/5 is also common (both Fibonacci Numbers), and
    5/8 also (you guessed it!)

    all getting closer and closer to the Golden Ratio.

    And that is why Fibonacci Numbers are very common in plants.
    1, 2, 3, 5, 8, 13, 21, ... etc occur in an amazing number of places.

    Here is a daisy with 21 petals
    (but expect a few more or less, because
    some may have dropped off or be just growing)

    But we don't see this in all plants, as nature has many different methods of survival.

    Golden Angle

    So far we have been talking about "turns" (full rotations).

    The equivalent of 0.61803... rotations is 222.4922... degrees, or about 222.5.

    In the other direction it is about 137.5, called the "Golden Angle".

    So, next time you are walking in the garden, look for the Golden Angle,
    and count petals and leaves to find Fibonacci Numbers,
    and discover how clever the plants are ... !


    Why don't you go into the garden or park right now, and start counting leaves and petals, and measuring rotations to see what you find.

    You can write your results on this form:

    Plant Name or Description:

    Do the Leaves Grow in Spirals? Y / N
    Count a group of Leaves:
    How many leaves (a) ?
    How many full rotations (b) ?
    Rotation per leaf (b/a) :
    Rotation Angle (360 b/a) :

    Are There Flowers? Y / N
    How many petals on Flower 1:
    Flower 2:
    Flower 3:

    (But remember, nature has its own rules, and it does not have to follow mathematical patterns,
    but when it does it is awesome to see.)

    * Notes About the Animation

    Sunflower seeds grow from the center outwards, but on the animation I found it easier to draw the younger seeds first and add on the older ones.

    The animation should continue longer to be the same as the sunflower - this would result in 55 clockwise spirals and 34 counterclockwise spirals (successive Fibonacci Numbers). I just didn't want it to take too long.

    The spirals are not programmed into it - they occur naturally as a result of trying to place the seeds as close to each other as possible while keeping them at the correct rotation.

    ◦Golden Ratio

    ◦Fibonacci Sequence
    ◦Irrational Numbers


    Polloi gar eisin kletoi oligoi de eklektoi - for many are called but few are chosen. Jesus made this statement in Matthew 22;14, after giving the parable of the marriage feast, in which the king invited his subjects to the marriage of his son, and one was found without a wedding garment. The king ordered he should be cast into outer darkness, after he had asked him why he was without a wedding garment.

    The wedding garment is the one we obtain by Christ, the Son of the King, However this guy had his own righteousness, just as many do today. We say Jesus has taken all our sins away, however, people believe in their own righteousness, and not in the righteousness of Christ, the one we obtain through him..


    The Bible says in Genesis 11:1, And the whole earth was of one language, and of one speech. They were of one Saphah, and of one Davar. Later, as they moved from the east, they made slime for mortar, and brick for stone.

    When Jesus met Peter, he first preached the word of God, Logos. Later, he gave Peter a specific word (Rhema) to cast the net by the side of the boat, to catch fish. Rhema and Logos are Greek words, used in the New Testament, Saphah and Davar are Hebrew words, used in the Old.

    As men moved from the east, they made slime for mortar, and brick for stone, in Genesis 11. Brick Levenah in Hebrew can be broken down into words meaning son of a woman, whereas stone Even, can be be broken down into two words, Av father, and Ben son, son of the father. So, as the children of men moved from their east, they left the son of the father, for the son of the woman. This is symbolic of leaving Christ the Son of the Father, for Satan.

    We must live by the complete word of God Logos and Rhema (Saphah and Davar). We must live by all the word of God, not doing this is to our disadvantage. We must not live by our thoughts, we must make a clear demarcation between our thoughts , and the word of God. Usually our thoughts seem to us, to be very real, and to be the fact, but we must make a demarcation between our thoughts, and the word of God.

    Usually our fears seem so real to us, this is part of our thoughts. The Bible says as far as the east is from the west, so has he separated our transgressions from us. The Christ of is that of the forgiveness of sins, the east. The garden was planted east of Eden, in the garden of Eden. The true Gospel, the true word of God, is that which preaches the total forgiveness of sins.


    Confession is derived from the Greek word Homologeo in the New Testament. The first part of the word Homolegeo, Homo, implies same, or of the same kind. The second is derived from Logos word.

    To confess means to say the same thing as God, or to agree with his word. The word of God says Jesus took all our sin away. It says this is the lamb of God that taketh away the sin (Harmatia, singular) of the world.

    The Bible also says if we say we have no sin we deceive ourselves, but if we.confess our sins he (God) is faithful and just to forgive us all our sins, and to cleanse us from all unrighteousness. So, many believe this portion of the Bible admonishes we should always confess our sins. Actually, as I mentioned before Jesus paid the whole price for sin, he carried all our sins.away.

    The Bible also says God will remember our sins and iniquities no more, when we believe in Christ. The confession the Bible refers to in the book of 1 John, where it says.if we confess our sins he is faithful and just to forgive us all our sins, and to cleanse us from all unrighteousness, is the confession of Christ as our Lord and Saviour. And that is what God does, he wipes all our sins away. The Bible says he is faithful and just to forgive us our sins, and to cleanse us from all unrighteousness.

    Most people do not like reading the book of Leviticus, because they say they find it boring, very boring. Perhaps, the book of Leviticus is probably one of the most significant in the Bible, since it reveals a lot about Christ, and his sacrifice for sin.

    The book of Leviticus tells us about the day of Atonement (Yom Kippur). On this day the High Priest performed the sacrifice for the whole nation of Israel (entering the Holy of Holies, the only sacrifice in which blood is carried into the Holy of Holies), confessing the sins of the whole nation. There was also the evening sacrifice, and the morning sacrifice.

    The sacrifice of the day of Atonement typifies the sacrifice of Jesus Christ, our High Priest. The High Priest must not enter the Holy of Holies with sin, in the Old Testament. If he did, he died. We can not enter heaven with sin, and we can only enter by Christ.

    The sacrifice of the evening and the morning typifies the other type of confession Christians make. Even though our sins have been forgiven, we sometimes have guilt when we commit sin. So, we confess our faults before the brethren, to be rid of guilt. The Bible says confess your faults one to another, so you might be healed.


    The tabernacle in the wilderness was oriented east to west. On the eastern part was the brazen alter. As you moved westward, you encountered the Holy Place, and lastly, the Holy of Holies.

    The garden was planted east, in the garden of Eden. The Bible says as the east is far from the west, so hath he separated our transgressions from us.

    Eden means pleasure, enjoyment. For those enlightened Eden also means the world, it means the church, it has something to do with the church.

    At first, I thought the Anti-Christ will come from the east of the world, since that is where the tree of the knowledge is (I might still be right, in a way). Actually, it will be a sort of marriage between the Anti-Christ and the False Prophet. One from the east, the other from the west

    The Bible say, he, talking about the Anti-Christ, will sit in yhe temple of God, making himself God. The throne of God is the west actually, in the Holy of Holies.

    The Anti-Christ will come from the most powerful nation on earth, probably, at that time.


    In the Old Covenant sin was only covered, never taken away. Jesus is the perfect sacrifice for sin.

    Many people think they believe Jesus, actually, they don't. If you believe in Jesus, you must believe he paid the price for all your sins, not just for some of your sins.

    If we would enter heaven, we must be without sin. So how can we be without sin, when we do sin. This means we must enter heaven by Jesus, through the righteousness of Jesus, and not by our own righteousness.

    The Bible says Jesus paid the price for all our sins, so he might deliver us from this present evil world. The word of God is a double edged sword, the Bible says we must rightly divide the word of God.

    The name Jesus implies he delivered us from our sin, if you do not believe he took all your sins away, you really do not believe Jesus.

    The problem people have is, they say, are you saying we can just keep on sinning, since Jesus took all our sins away? Who is saying that anyway? Nobody!

    The fact remains, don't you commit sin, even though you are trying to live "a sinless life," even as you "strive toward perfection." You still commit sin, so what is the difference between you, and the one who believes all is sins is paid for.

    The difference is heaven, where the throne of God dwells. The one who believes all his sins is paid for, will dwell with God for all eternity.


    Hebrews 10 King James Version (KJV)

    1 For the law having a shadow of good things to come, and not the very image of the things, can never with those sacrifices which they offered year by year continually make the comers thereunto perfect.

    2 For then would they not have ceased to be offered? because that the worshippers once purged should have had no more conscience of sins.

    3 But in those sacrifices there is a remembrance again made of sins every year.

    4 For it is not possible that the blood of bulls and of goats should take away sins.

    5 Wherefore when he cometh into the world, he saith, Sacrifice and offering thou wouldest not, but a body hast thou prepared me:

    6 In burnt offerings and sacrifices for sin thou hast had no pleasure.

    7 Then said I, Lo, I come (in the volume of the book it is written of me,) to do thy will, O God.

    8 Above when he said, Sacrifice and offering and burnt offerings and offering for sin thou wouldest not, neither hadst pleasure therein; which are offered by the law;

    9 Then said he, Lo, I come to do thy will, O God. He taketh away the first, that he may establish the second.

    10 By the which will we are sanctified through the offering of the body of Jesus Christ once for all.

    11 And every priest standeth daily ministering and offering oftentimes the same sacrifices, which can never take away sins:

    12 But this man, after he had offered one sacrifice for sins for ever, sat down on the right hand of God;

    13 From henceforth expecting till his enemies be made his footstool.

    14 For by one offering he hath perfected for ever them that are sanctified.

    15 Whereof the Holy Ghost also is a witness to us: for after that he had said before,

    16 This is the covenant that I will make with them after those days, saith the Lord, I will put my laws into their hearts, and in their minds will I write them;

    17 And their sins and iniquities will I remember no more.

    18 Now where remission of these is, there is no more offering for sin.

    19 Having therefore, brethren, boldness to enter into the holiest by the blood of Jesus,

    20 By a new and living way, which he hath consecrated for us, through the veil, that is to say, his flesh;

    21 And having an high priest over the house of God;

    22 Let us draw near with a true heart in full assurance of faith, having our hearts sprinkled from an evil conscience, and our bodies washed with pure water.

    23 Let us hold fast the profession of our faith without wavering; (for he is faithful that promised;)

    24 And let us consider one another to provoke unto love and to good works:

    25 Not forsaking the assembling of ourselves together, as the manner of some is; but exhorting one another: and so much the more, as ye see the day approaching.

    26 For if we sin wilfully after that we have received the knowledge of the truth, there remaineth no more sacrifice for sins,

    27 But a certain fearful looking for of judgment and fiery indignation, which shall devour the adversaries.

    28 He that despised Moses' law died without mercy under two or three witnesses:

    29 Of how much sorer punishment, suppose ye, shall he be thought worthy, who hath trodden under foot the Son of God, and hath counted the blood of the covenant, wherewith he was sanctified, an unholy thing, and hath done despite unto the Spirit of grace?

    30 For we know him that hath said, Vengeance belongeth unto me, I will recompense, saith the Lord. And again, The Lord shall judge his people.

    31 It is a fearful thing to fall into the hands of the living God.

    32 But call to remembrance the former days, in which, after ye were illuminated, ye endured a great fight of afflictions;

    33 Partly, whilst ye were made a gazingstock both by reproaches and afflictions; and partly, whilst ye became companions of them that were so used.

    34 For ye had compassion of me in my bonds, and took joyfully the spoiling of your goods, knowing in yourselves that ye have in heaven a better and an enduring substance.

    35 Cast not away therefore your confidence, which hath great recompence of reward.

    36 For ye have need of patience, that, after ye have done the will of God, ye might receive the promise.

    37 For yet a little while, and he that shall come will come, and will not tarry.

    38 Now the just shall live by faith: but if any man draw back, my soul shall have no pleasure in him.

    39 But we are not of them who draw back unto perdition; but of them that believe to the saving of the soul.

    The Bible says without the shedding of blood there is no remission of sin. However, most people think it is when they ask, when "they confess their sins" God forgives them. If there is no shedding of bloood, there is no remission of sin.

    Jesus' blood was shed once. The Bible calls the sacrifice of Jesus a perfect sacrifice. Just after this passage, Hebrews 10, the next chapter, Hebrews 11, talks about faith.

    Do you believe Jesus died for all your sins? Do you believe his sacrifice is perfect? So why do you always ask God should forgive you, when he says your sins and iniquities will I remember no more.

    God promises to give us his Holy Spirit, when we believe. So why do you think you can take care of yout sins all by yourself?

    The Bible says if we sin willfully, there is no more sacrifice for sin. What is the willful sin? Does it mean you would no longer commit sin, after you have accepted Jesus? The willful sin is if you have accepted Christ, afterward, you reject him. The Bible says there is no man that liveth and sinneth not.


    Zechariah 4:1-7King James Version (KJV)

    1 And the angel that talked with me came again, and waked me, as a man that is wakened out of his sleep.

    2 And said unto me, What seest thou? And I said, I have looked, and behold a candlestick all of gold, with a bowl upon the top of it, and his seven lamps thereon, and seven pipes to the seven lamps, which are upon the top thereof:

    3 And two olive trees by it, one upon the right side of the bowl, and the other upon the left side thereof.

    4 So I answered and spake to the angel that talked with me, saying, What are these, my lord?

    5 Then the angel that talked with me answered and said unto me, Knowest thou not what these be? And I said, No, my lord.

    6 Then he answered and spake unto me, saying, This is the word of the Lord unto Zerubbabel, saying, Not by might, nor by power, but by my spirit, saith the Lord of hosts.

    7 Who art thou, O great mountain? before Zerubbabel thou shalt become a plain: and he shall bring forth the headstone thereof with shoutings, crying, Grace, grace unto it.

    We try not to sin, most people try not to commit sin. People try to lead a "sinless life," if there is any such thing as a "sinless life."

    People say they are striving toward perfection. Actially, if we believe in Christ, if we believe our sins are forgiven, we are perfect. The Bible says not by power, not by might, but by my Spirit, says the LORD.

    Which one would you believe? The Bible says you are perfect, even though you commit sin, if you believe in Jesus (like Job, Zachariah and Elizabeth). Or, you believe you are not perfect when you commit sin, even though you believe in Jesus.

    We can not overcome sin by trying to keep the law, or by trying not to sin. We can only overcome sin, by believing in Jesus, that he took all our sins away. Just as in the case of Zerubbabel, all our righteousness is like filthy rags.

    Grace comes by believing in Jesus. We believe Jesus took our sins away. Grace is what we need? Yes grace is what we need, the grace of God.


    Galatians 2:14-21King James Version (KJV)

    14 But when I saw that they walked not uprightly according to the truth of the gospel, I said unto Peter before them all, If thou, being a Jew, livest after the manner of Gentiles, and not as do the Jews, why compellest thou the Gentiles to live as do the Jews?

    15 We who are Jews by nature, and not sinners of the Gentiles,

    16 Knowing that a man is not justified by the works of the law, but by the faith of Jesus Christ, even we have believed in Jesus Christ, that we might be justified by the faith of Christ, and not by the works of the law: for by the works of the law shall no flesh be justifiged.

    17 But if, while we seek to be justified by Christ, we ourselves also are found sinners, is therefore Christ the minister of sin? God forbid.

    18 For if I build again the things which I destroyed, I make myself a transgressor.

    19 For I through the law am dead to the law, that I might live unto God.

    20 I am crucified with Christ: nevertheless I live; yet not I, but Christ liveth in me: and the life which I now live in the flesh I live by the faith of the Son of God, who loved me, and gave himself for me.

    21 I do not frustrate the grace of God: for if righteousness come by the law, then Christ is dead in vain.

    Verse 17 of the passage above says if we try to be justified by Christ, we too are found to be sinners, is Christ the minister of sin? The question remains - who is a sinner, and who is the minister of sin? A sinner is not someone who does not commit sin, we all do commit sin. A sinner is someone who does not believe God has forgiven his sins completely. The minister of sin is when we believe Satan, the minister of sin, rather than believe God.

    The Bible says somewhere return unto me for I have pardoned your sins. So, which comes first, is it the returning, or is it the pardoning? Also in the story of the prodigal son, did the father forgive him when he returned home, or he forgave him before that?

    The pardoning comes before the returning, and the father actually forgave the son when he left him initially, not when he returned home. So also, the Bible says while we were yet sinners Christ died for us. The righteous are those who believe the word of God, they are those who believe they have been justified with Christ, by faith, and not according to the law.

    When we believe in Jesus Christ, God gives us his Holy Spirit. Who wouldn't want the the Holy Spirit be the guide in life. Nevetheless, many try to overcome sin, all by themselves.


    The priesthood of Jesus is the Melchizedek priesthood. Under the Aaronic priesthood, we sin and we perpetually ask for forgiveness, in the Old Testament. Under the New, we are completely washed, and our sins are taken away, we are completely forgiven. It is a wide concept, and I still have much to say.

    The Bible says by one man we were made sinners and by one man (Christ Jesus) we will be made righteous. We always think we can take care of our sins by ourselves. I have not seen anyone who was committing sin before accepting Jesus, who ceased from doing so, after accepting Jesus.

    So if we commit sin, even after we accept the Christ, why do we think Christ died for our past sins only? Many people do not have any doubts about that, about their past sins, that Jesus paid the penalty for that one.

    The Bible says should we continue in sin and say the grace of God should abound? Two things - continue in sin, the grace of God should abound. What is the grace of God. The grace of God is, Jesus paid the penalty for all our sins, we do not have to pay the penalty again. We continue in sin, if we do not believe this.

    Jesus came to die for the sins of the whole world, we either choose to believe this, or we choose not to believe. The Bible says the law came by Moses, but grace and truth came by Jesus.

    The disciples of Jesus asked him to teach them to pray " as John taught his disciples to pray." Jesus told them to pray in part like this - Father forgive us our trespasses as we forgive those who trespass against us - and at that period in time Jesus had not shed his blood on the cross. Later as he was about to die he boldly proclaimed, this is my blood of the new covenant shed for the remission of sin. Now, I would ask you all to do a study on the word remission, especially as it is found in the original Greek

    Also Jesus told the woman caught in adultery, go and sin no more. The law brought the woman to Jesus. She should have been condemned to death, however Jesus set het free, but she had not met with Jesus. Jesus said go and sin no more, he gave her the law, so she would know she could not keep it, so she would eventually discover him. Because Jesus told the woman go and sin no more, so now people try to lead "a sinless life." There is nothing like "a sinless life" in reality. The Bible says there is no man that liveth and sinneth not.

    Perfection is not attained, perfection is obtained, in Christ. Most people say they are striving toward perfection, but when we believe in Christ, we obtain perfection from the LORD.

    We have many examples in the Bible. Job was said to be perfect, in the Bible. Zachariah and Elizabeth were perfect, they were said to be upright and blameless before the LORD.

    If I asked you a question, if you commit sin, even after you accept the Christ, does that make you less perfect before God? If you say no to this I will tell you one thing - you need to check your faith in Christ.

    Job, Zachariah and Elizabeth were said to be perfect in the Bible, because they did not have their own righteousness but the righteousness of the LORD, just ws we do the righteousness of our Lord Jesus. They all used to commit sin. We are no more sinners, that is why Christ died for us, though we still commit sin - by one man we were made sinners, and by one man we will be made righteous.

    This is why the Bible talks of faith. God says Jesus took all our sins away, we should believe this. If God says something is white, though to all your sesnses it looks black, which one would you choose to believe?


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